$\mathcal{G}_{1}\perp\\!\\!\\!\perp_{\mathcal{H}^{\prime}}\mathcal{G}_{2}$ for
any $\sigma$-algebra $\mathcal{H}^{\prime}\supseteq\mathcal{H}$ such that
$\mathcal{H}^{\prime}\subseteq\mathcal{G}_{1}$;

  3. (c)

$\mathcal{G}_{1}\perp\\!\\!\\!\perp_{\mathcal{H}^{\prime}}\mathcal{G}_{2}$ for
any $\sigma$-algebra $\mathcal{H}^{\prime}\supseteq\mathcal{H}$ of the form
$\mathcal{H}^{\prime}=\mathcal{H}^{\prime}_{1}\vee\mathcal{H}_{2}^{\prime},$
where $\mathcal{H}^{\prime}_{1},\mathcal{H}^{\prime}_{2}$ are
$\sigma$-algebras satisfying
$\mathcal{H}_{1}^{\prime}\subseteq\mathcal{G}_{1}\vee\mathcal{H}$ and
$\mathcal{H}_{2}^{\prime}\subseteq\mathcal{G}_{2}\vee\mathcal{H}$.

###### Proof.

Part a is [21, Corollary 8.11(i)]; combining it with
$\mathcal{G}_{1}=\mathcal{G}_{1}\vee\mathcal{H}^{\prime}$ and
${\mathcal{H}^{\prime}=\mathcal{H}\vee\mathcal{H}^{\prime}}$ yields b. To
prove (c), first note that
$\mathcal{G}_{1}\vee\mathcal{H}\perp\\!\\!\\!\perp_{\mathcal{H}\vee\mathcal{H}_{1}^{\prime}}\mathcal{G}_{2}\vee\mathcal{H}$
by parts a and b. Applying a again, we find

$\mathcal{G}_{1}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}\perp\\!\\!\\!\perp_{\mathcal{H}\vee\mathcal{H}_{1}^{\prime}}\mathcal{G}_{2}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}.$

Since
$\mathcal{H}\vee\mathcal{H}_{1}^{\prime}\subseteq\mathcal{H}\vee\mathcal{H}^{\prime}=\mathcal{H}^{\prime}=\mathcal{H}_{2}^{\prime}\vee\mathcal{H}_{1}^{\prime}\subseteq\mathcal{G}_{2}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}$,
part b yields

$\mathcal{G}_{1}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}\perp\\!\\!\\!\perp_{\mathcal{H}^{\prime}}\mathcal{G}_{2}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime},$

which proves c since
$\mathcal{G}_{1}\subseteq\mathcal{G}_{1}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}$
and
$\mathcal{G}_{2}\subseteq\mathcal{G}_{2}\vee\mathcal{H}\vee\mathcal{H}_{1}^{\prime}$.
∎

###  B.2. Facts regarding Assumption 4.2(i)

######  Lemma B.3.

Let Assumption 4.1 be satisfied, i.e., suppose the linear operator
$-A\colon\mathsf{D}(A)\subseteq U\to U$ on the separable real Hilbert space
$U$ generates an exponentially stable $C_{0}$-semigroup $(S(t))_{t\geq 0}$ of
bounded linear operators on $U$. If $Q\in\mathscr{L}^{+}(U)$ and
$\gamma_{0}\in(-\infty,\nicefrac{{1}}{{2}}]$, then

$\int_{0}^{\infty}\|t^{\gamma_{0}-1}S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}t=\infty,$

that is, Assumption 4.2(i) cannot hold for
$\gamma_{0}\in(-\infty,\nicefrac{{1}}{{2}}]$.

###### Proof.

Fix some $x\in U$ with $\|x\|_{U}=1$. Then
$\|S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}\geq\|S(t)Q^{\frac{1}{2}}x\|_{U}$
for all $t\in[0,\infty)$. Since $t\mapsto S(t)Q^{\frac{1}{2}}x$ is continuous
at zero and $S(0)Q^{\frac{1}{2}}x=Q^{\frac{1}{2}}x$, we can choose
$\delta\in(0,\infty)$ so small that
$\|S(t)Q^{\frac{1}{2}}x\|_{U}\geq\frac{1}{2}\|Q^{\frac{1}{2}}x\|_{U}$ for all
$t\in[0,\delta]$. If $\gamma_{0}\in(-\infty,\nicefrac{{1}}{{2}}]$, we then
obtain

$\int_{0}^{\infty}\|t^{\gamma_{0}-1}S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}t\geq\frac{1}{2}\|Q^{\frac{1}{2}}x\|_{U}\int_{0}^{\delta}t^{2(\gamma_{0}-1)}\mathop{}\\!\mathrm{d}t=\infty.\qed$

######  Lemma B.4.

Let Assumption 4.1 be satisfied. If Assumptions 4.2(i) holds for some
$\gamma_{0}\in(\nicefrac{{1}}{{2}},\infty)$, then it also holds for all
$\gamma^{\prime}\in[\gamma_{0},\infty)$.

###### Proof.

The change of variables $\tau:=t/2$, the semigroup property and (4.2) yield

$\displaystyle\int_{0}^{\infty}\|t^{\gamma^{\prime}-1}S(t)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq
2^{2\gamma^{\prime}-1}M_{0}^{2}\int_{0}^{\infty}e^{-2w\tau}\|\tau^{\gamma^{\prime}-1}S(\tau)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}\tau.$

For the latter integral, we split up the domain of integration and estimate
each of the resulting integrands to find

$\displaystyle\int_{0}^{\infty}$ $\displaystyle
e^{-2w\tau}\|\tau^{\gamma^{\prime}-1}S(\tau)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}\tau=\sum_{k=1}^{\infty}\int_{k-1}^{k}e^{-2w\tau}\|\tau^{\gamma^{\prime}-1}S(\tau)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}\tau$
$\displaystyle\leq\sum_{k=1}^{\infty}e^{-2w(k-1)}k^{2(\gamma^{\prime}-\gamma_{0})}\int_{0}^{\infty}\|\tau^{\gamma_{0}-1}S(\tau)Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(U)}^{2}\mathop{}\\!\mathrm{d}\tau<\infty,$

where the series converges since
$|e^{-2wk}k^{2(\gamma^{\prime}-\gamma_{0})}|^{\frac{1}{k}}\to e^{-2w}<1$ as
$k\to\infty$. ∎

###  B.3. Filtrations indexed by the real line

The proof of Proposition B.6 below elaborates on [2, Example 3.6]. First we
state and prove a lemma which shows that collections of random variables are
generated by a $\pi$-system consisting of finite products of bounded, real-
valued random variables.

######  Lemma B.5.

If $(X_{i})_{i\in\mathcal{I}}$ is a collection of $U$-valued random variables,
then

$\sigma(X_{i}:i\in\mathcal{I})=\sigma\left(\prod\nolimits_{i\in\mathcal{J}}Z_{i}:\mathcal{J}\subseteq\mathcal{I}\textup{
finite},\,Z_{i}\in B_{b}(\Omega,\sigma(X_{i})),\forall
i\in\mathcal{J}\\!\right)\\!.$ (B.2)

###### Proof.

The “$\supseteq$” inclusion holds because any $Z=\prod_{i\in\mathcal{J}}Z_{i}$
is measurable with respect to
$\bigvee_{i\in\mathcal{J}}\sigma(X_{i})\subseteq\bigvee_{i\in\mathcal{I}}\sigma(X_{i})$
for any $\mathcal{J}\subseteq\mathcal{I}$. For the converse inclusion, note
that the join of a family of $\sigma$-algebras
$(\mathcal{A}_{i})_{i\in\mathcal{I}}$ has the following general property:

$\bigvee_{i\in\mathcal{I}}\mathcal{A}_{i}=\sigma\left(\bigcap\nolimits_{i\in\mathcal{J}}A_{i}:\mathcal{J}\subseteq\mathcal{I}\text{
finite},\,A_{i}\in\mathcal{A}_{i}\,\,\forall i\in\mathcal{J}\right)\\!.$

Thus, by definition of $\sigma(X_{i})$, we obtain (see also [40, Chapter 0,
Theorem 2.3])

$\sigma(X_{i}:i\in\mathcal{I})=\sigma\left(\bigcap\nolimits_{i\in\mathcal{J}}X_{i}^{-1}[B_{i}]:\mathcal{J}\subseteq\mathcal{I}\text{
finite},\,B_{i}\in\mathcal{B}(U)\,\,\forall i\in\mathcal{J}\right)\\!.$ (B.3)

Since
$\bigcap_{i\in\mathcal{J}}X_{i}^{-1}[B_{i}]=(\prod_{i\in\mathcal{J}}\mathbf{1}_{B_{i}}(X_{i}))^{-1}[\\{1\\}]$
for any finite $\\{B_{i}:i\in\mathcal{J}\\}\subseteq\mathcal{B}(U)$, we find
that the right-hand side of (B.3) is contained in that of (B.2). ∎

######  Proposition B.6.

Let $(W^{Q}(t))_{t\in\mathbb{R}}$ be a $U$-valued two-sided $Q$-Wiener process
with covariance operator $Q\in\mathscr{L}_{1}^{+}(U)$, see Subsection 2.2. The
filtration $(\mathcal{F}^{\delta W^{Q}}_{t})_{t\in\mathbb{R}}$ defined in
(2.3) is right-continuous, i.e., we have $\mathcal{F}_{t+}^{\delta
W^{Q}}:=\bigcap_{\varepsilon>0}\mathcal{F}_{t+\varepsilon}^{\delta
W^{Q}}=\mathcal{F}_{t}^{\delta W^{Q}}$ for all $t\in\mathbb{R}$.

###### Proof.

Fix $t\in\mathbb{R}$. It suffices to prove that

$\mathbb{E}[Z\mid\mathcal{F}^{\delta
W^{Q}}_{t+}]=\mathbb{E}[Z\mid\mathcal{F}^{\delta
W^{Q}}_{t}],\quad\mathbb{P}\text{-a.s.},$ (B.4)

for all $Z\in B_{b}(\Omega,\mathcal{F}_{\infty}^{\delta W^{Q}})$, where
$\mathcal{F}_{\infty}^{\delta W^{Q}}:=\sigma(W_{s,u}:s\leq u)$, using the
shorthand notation $W_{s,u}=W^{Q}(u)-W^{Q}(s)$. Indeed, given
$A\in\mathcal{F}_{t+}^{\delta W^{Q}}\subseteq\mathcal{F}_{\infty}^{\delta
W^{Q}}$, equation (B.4) would imply that $\mathbf{1}_{A}$ is $\mathbb{P}$-a.s.
equal to the $\mathcal{F}_{t}^{\delta W^{Q}}$-measurable random variable
$\mathbb{E}[\mathbf{1}_{A}\mid\mathcal{F}^{\delta W^{Q}}_{t}]$, and thus
$A\in\mathcal{F}_{t}^{\delta W^{Q}}$ by the completeness of
$(\mathcal{F}_{t}^{\delta W^{Q}})_{t\in\mathbb{R}}$. This yields
$\mathcal{F}_{t+}^{\delta W^{Q}}\subseteq\mathcal{F}_{t}^{\delta W^{Q}}$.

Let $\mathscr{H}$ denote the linear space of bounded random variables
$Z\colon\Omega\to\mathbb{R}$ for which (B.4) holds; we want to show that
$B_{b}(\Omega,\mathcal{F}_{\infty}^{\delta W^{Q}})\subseteq\mathscr{H}$ using
the monotone class theorem [40, Chapter 0, Theorem 2.2]. Let $\mathscr{C}$
denote the set of random variables $Z\colon\Omega\to\mathbb{R}$ which can be
written as a product of two bounded random variables
$Z_{1},Z_{2}\colon\Omega\to\mathbb{R}$, where $Z_{1}$ is
$\mathcal{F}_{t}^{\delta W^{Q}}$-measurable and $Z_{2}$ is measurable with
respect to $\sigma(W_{s,u}:t+\varepsilon\leq s\leq
u)\vee\sigma(\mathcal{N}_{\mathbb{P}})$ for some $\varepsilon>0$. We first
observe that $\mathscr{C}\subseteq\mathscr{H}$. Indeed, for all
$Z\in\mathcal{K}$ with $Z=Z_{1}Z_{2}$ as above,

$\mathbb{E}[Z\mid\mathcal{F}_{t+}^{\delta
W^{Q}}]=\mathbb{E}[Z_{1}Z_{2}\mid\mathcal{F}_{t+}^{\delta
W^{Q}}]=Z_{1}\mathbb{E}[Z_{2}]=\mathbb{E}[Z_{1}Z_{2}\mid\mathcal{F}_{t}^{\delta
W^{Q}}]=\mathbb{E}[Z\mid\mathcal{F}_{t}^{\delta W^{Q}}],$

$\mathbb{P}$-a.s., since $Z_{1}$ is measurable with respect to
$\mathcal{F}_{t}^{\delta W^{Q}}\subseteq\mathcal{F}_{t+}^{\delta W^{Q}}$,
while $Z_{2}$ is independent of $\mathcal{F}_{t+}^{\delta
W^{Q}}\supseteq\mathcal{F}_{t}^{\delta W^{Q}}$ by (WP3′). We note moreover
that $\mathscr{C}$ is closed under pointwise multiplication and that
$\mathscr{H}$ contains all constant functions and satisfies $Z\in\mathscr{H}$
whenever $(Z_{n})_{n\in\mathbb{N}}\subseteq\mathscr{H}$ is such that $0\leq
Z_{n}\uparrow Z$ by conditional monotone convergence. Thus the conditions of
the monotone class theorem are indeed satisfied, hence
$B_{b}(\Omega,\sigma(\mathscr{C}))\subseteq\mathscr{H}$ and it remains to show
$\sigma(\mathscr{C})=\mathcal{F}^{\delta W^{Q}}_{\infty}$.

The inclusion $\sigma(\mathscr{C})\subseteq\mathcal{F}^{\delta
W^{Q}}_{\infty}$ follows from the fact that any random variables $Z_{1}$ and
$Z_{2}$ as above are both $\mathcal{F}^{\delta W^{Q}}_{\infty}$-measurable,
implying the same for $Z$. For the converse inclusion, we first split
$\mathcal{F}_{\infty}^{\delta W^{Q}}$ at time $t$ as follows:

$\mathcal{F}^{\delta W^{Q}}_{\infty}=\sigma(W_{s,u}:s\leq
u)=\mathcal{F}_{t}^{\delta W^{Q}}\vee\sigma(W_{s,u}:t\leq s\leq u).$ (B.5)

Since $W^{Q}$ is (right-)continuous, $\mathbb{P}$-a.s., we have in fact

$\mathcal{F}^{\delta W^{Q}}_{\infty}=\mathcal{F}_{t}^{\delta
W^{Q}}\vee\sigma(W_{s,u}:t<s\leq u),$

which can also be written as

$\mathcal{F}_{\infty}^{\delta
W^{Q}}=\sigma(W_{s,u}:(s,u)\in\mathcal{I})\vee\sigma(\mathcal{N}_{\mathbb{P}}),$

where $\mathcal{I}:=\mathcal{I}_{1}\cup\mathcal{I}_{2}$ with
$\mathcal{I}_{1}:=\\{(s,u):s\leq u\leq t\\}$ and
$\mathcal{I}_{2}:=\\{(s,u):t<s\leq u\\}$. An application of Lemma B.5 to the
above equation shows that $\mathcal{F}_{\infty}^{\delta W^{Q}}$ is equal to

$\sigma\biggl{(}\prod\nolimits_{(s,u)\in\mathcal{J}}Y_{(s,u)}:\mathcal{J}\subseteq\mathcal{I}\textup{
finite},\,Y_{(s,u)}\in
B_{b}(\Omega,\sigma(W_{s,u}))\,\,\forall(s,u)\in\mathcal{J}\\!\biggr{)}\\!\vee\sigma(\mathcal{N}_{\mathbb{P}}).$

Consider $Z:=\prod_{(s,u)\in\mathcal{J}}Y_{(s,u)}$ for some finite
$\mathcal{J}\subseteq\mathcal{I}$. Define
$Z_{1}:=\prod_{(s,u)\in\mathcal{J}\cap\mathcal{I}_{1}}Y_{(s,u)}$ and
$Z_{2}:=\prod_{(s,u)\in\mathcal{J}\cap\mathcal{I}_{2}}Y_{(s,u)}$. Then we have
$Z=Z_{1}Z_{2}$, with $Z_{1}$ and $Z_{2}$ as in the definition of
$\mathscr{C}$, finishing the proof of the identity
$\mathcal{F}_{\infty}^{\delta W^{Q}}=\sigma(\mathscr{C})$. ∎

######  Proposition B.7.

A process $(W^{Q}(t))_{t\in\mathbb{R}}$ satisfying (WP1) cannot be a
martingale with respect to any filtration
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$.

###### Proof.

Suppose that $(W^{Q}(t))_{t\in\mathbb{R}}$ is a martingale with respect to
some filtration $(\mathcal{F})_{t\in\mathbb{R}}$. Then the same holds for the
real-valued process $W_{h}^{Q}(t):=\langle W^{Q}(t),h\rangle_{U}$, where we
choose $h\in U$ such that $\langle Qh,h\rangle_{U}^{2}>0$ to ensure that
$(W_{h}^{Q}(t))_{t\in\mathbb{R}}$ has nontrivial increments. In particular,
$(W^{Q}_{h}(-n))_{n\in\mathbb{N}}$ is a backward martingale with respect to
$(\mathcal{F}_{-n})_{n\in\mathbb{N}}$, implying that it converges
$\mathbb{P}$-a.s. and in $L^{1}(\Omega)$ as $n\to\infty$ by the backward
martingale convergence theorem [15, Section 12.7, Theorem 4]. But this
contradicts (WP1), since $(W^{Q}_{h}(-n))_{n\in\mathbb{N}}$ cannot be a Cauchy
sequence in $L^{1}(\Omega)$ as the process has (non-trivial) stationary
increments. ∎

###  B.4. Mean-square differentiability of stochastic convolutions

The following lemma regarding mean-square continuity and differentiation under
the integral sign generalizes [24, Propositions 3.18 and 3.21] to stochastic
convolutions with respect to a two-sided Wiener process.

######  Lemma B.8.

Let $t_{0}\in[-\infty,\infty)$ be such that $\Psi(\,\cdot\,)Q^{\frac{1}{2}}\in
L^{2}(0,\infty;\mathscr{L}_{2}(U))$ and set $\mathbb{T}:=[t_{0},\infty)$ if
$t_{0}\in\mathbb{R}$ or $\mathbb{T}:=\mathbb{R}$ if $t_{0}=-\infty$. Then the
stochastic convolution
$(\int_{t_{0}}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s))_{t\in\mathbb{T}}$
is mean-square continuous.

If $\Psi(\,\cdot\,)Q^{\frac{1}{2}}\in
H^{1}_{0,\\{0\\}}(0,\infty;\mathscr{L}_{2}(U))$, then
$(\int_{t_{0}}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s))_{t\in\mathbb{T}}$
is mean-square differentiable on $\mathbb{T}$ with

$\displaystyle\forall
t\in\mathbb{T}:\quad\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\int_{t_{0}}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s)=\int_{t_{0}}^{t}\partial_{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad\mathbb{P}\text{-a.s.}$
(B.6)

###### Proof.

Adapting the proof of [24, Proposition 3.18] amounts to establishing that

$\lim_{h\to
0}\int_{0\vee(-h)}^{\infty}\|[\Psi(u+h)-\Psi(u)]Q^{\frac{1}{2}}\|^{2}_{\mathscr{L}_{2}(U)}\mathop{}\\!\mathrm{d}u=0,$

which holds by the continuity of translations in
$L^{2}(0,\infty;\mathscr{L}_{2}(U))$, see for instance [24, Lemma A.4] for
this specific form. Similarly, for the mean-square differentiability we note
that compared to the proof of [24, Proposition 3.21], only the terms
$I_{1}^{h^{\pm}}$ change when passing from $t_{0}=0$ to a general $t_{0}$. For
$t\in\mathbb{T}$ and $h\in(0,\infty)$, we have

$I^{h^{+}}_{1}=\int_{t_{0}}^{t}\biggl{[}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{]}\mathop{}\\!\mathrm{d}W^{Q}(s),$
(B.7)

whereas for $t\in(t_{0},\infty)$ and $h\in[t_{0}-t,0)$ (or $h\in(-\infty,0)$
if $t_{0}=-\infty$),

$I^{h^{-}}_{1}=\int_{t_{0}}^{t+h}\biggl{[}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{]}\mathop{}\\!\mathrm{d}W^{Q}(s).$
(B.8)

Using the Itô isometry (2.2) and applying the change of variables $u:=t-s$:

$\displaystyle\bigl{\|}I^{h^{\pm}}_{1}\bigr{\|}^{2}_{L^{2}(\Omega;U)}=\int_{0\vee(-h)}^{t-t_{0}}\,\biggl{\|}\biggl{[}\frac{\Psi(u+h)-\Psi(u)}{h}-\Psi^{\prime}(u)\biggr{]}Q^{\frac{1}{2}}\biggr{\|}^{2}_{\mathscr{L}_{2}(U)}\mathop{}\\!\mathrm{d}u.$
(B.9)

The proof is finished upon noting that replacing $t$ by $t-t_{0}$ does not
affect the remaining step of the argument. ∎

##  Appendix C Fractional powers of the parabolic operator

Let $A\colon\mathsf{D}(A)\subseteq U\to U$ be a linear operator on a real
Hilbert space $U$.

######  Definition C.1.

Under Assumption 4.1, we define the negative fractional power operator
$A^{-\alpha}$ of order $\alpha\in(0,\infty)$ as the $\mathscr{L}(U)$-valued
Bochner integral

$A^{-\alpha}:=\frac{1}{\Gamma(\alpha)}\int_{0}^{\infty}t^{\alpha-1}S(t)\mathop{}\\!\mathrm{d}t.$
(C.1)

Then $A^{-\alpha}$ is injective and we define
$A^{\alpha}x:=(A^{-\alpha})^{-1}x$ for
$x\in\mathsf{D}(A^{\alpha}):=\mathsf{R}(A^{-\alpha})$. For $\alpha=0$ we set
$A^{0}:=\operatorname{Id}_{U}$.

See [37, Section 2.6] for more details regarding this definition of fractional
powers.

######  Lemma C.2.

Let $(S,\mathscr{A},\mu)$ be a measure space such that
$L^{2}(\Sigma;\mathbb{R})$ is nontrivial and consider the linear operator
$\mathcal{A}_{S}$ on $L^{2}(S;U)$ defined by (4.8). Then the following
statements hold:

  1. (a)

$\mathcal{A}_{S}\in\mathscr{L}(L^{2}(S;U))$ if and only if
$A\in\mathscr{L}(U)$, with equal operator norms;

  2. (b)

$\mathcal{A}_{S}$ is closed if and only if $A$ is;

  3. (c)

If $A$ generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on $U$, then
$\mathcal{A}_{S}$ generates the $C_{0}$-semigroup $(\mathcal{T}_{S}(t))_{t\geq
0}$ on $L^{2}(S;U)$.

###### Proof.

For (a)–(b) one can argue as in the proof of [24, Lemma A.1], replacing
$T^{-\frac{1}{2}}\mathbf{1}_{(0,T)}$ by another function with unit
$L^{2}(S;\mathbb{R})$-norm. The proof of (c) is entirely analogous to that of
[24, Proposition A.3]. ∎

The next two statements and their proofs are analogous to Propositions A.5 and
3.2 of [24], respectively.

######  Proposition C.3.

For $t\in\mathbb{R}$, let the operator
$\mathcal{T}(t)\in\mathscr{L}(L^{2}(\mathbb{R};U))$ be defined by

$\mathcal{T}(t)f:=f(\,\cdot\,-t),\quad f\in L^{2}(\mathbb{R};U).$ (C.2)

The family $(\mathcal{T}(t))_{t\in\mathbb{R}}$ is a $C_{0}$-group whose
infinitesimal generator is given by $-\partial_{t}$, where $\partial_{t}$ is
the Bochner–Sobolev weak derivative on
$\mathsf{D}(\partial_{t})=H^{1}(\mathbb{R};U)$.

######  Proposition C.4.

Suppose that Assumption 4.1 holds. The closure
$\mathcal{B}:=\overline{\partial_{t}+\mathcal{A}_{\mathbb{R}}}$ of the sum
operator $\partial_{t}+\mathcal{A}_{\mathbb{R}}$ exists and $-\mathcal{B}$
generates the $C_{0}$-semigroup
$(\mathcal{S}_{\mathbb{R}}(t)\mathcal{T}(t))_{t\geq 0}$ on
$L^{2}(\mathbb{R};U)$, which satisfies

$\forall
t\in\mathbb{R}\colon\quad\|\mathcal{S}_{\mathbb{R}}(t)\mathcal{T}(t)\|_{L^{2}(\mathbb{R};U)}=\|\mathcal{T}(t)\mathcal{S}_{\mathbb{R}}(t)\|_{L^{2}(\mathbb{R};U)}=\|S(t)\|_{\mathscr{L}(U)},$

where $(\mathcal{S}_{\mathbb{R}}(t))_{t\geq 0}$ and $(\mathcal{T}(t))_{t\geq
0}$ are defined as in (4.8) and (C.2), respectively.

We thus find that $(\mathcal{S}_{\mathbb{R}}(t)\mathcal{T}(t))_{t\geq 0}$
inherits the exponential stability of $(S(t))_{t\geq 0}$, so that fractional
powers of $\mathcal{B}$ can be defined using Definition C.1. Therefore, under
Assumption 4.1, equation (C.1) implies

$\mathcal{B}^{-\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}r^{\gamma-1}\mathcal{S}_{\mathbb{R}}(r)\mathcal{T}(r)f(t)\mathop{}\\!\mathrm{d}r=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}r^{\gamma-1}S(r)f(t-r)\mathop{}\\!\mathrm{d}r$

for $\gamma\in(0,\infty),f\in L^{2}(\mathbb{R};U)$ and almost all
$t\in\mathbb{R}$. We conclude that
$\mathcal{B}^{-\gamma}=\mathfrak{I}^{\gamma}$ for all $\gamma\in[0,\infty)$,
where the latter is defined by (4.5).

## References

  * [1] S. E. Alexeeff, D. Nychka, S. R. Sain, and C. Tebaldi, Emulating mean patterns and variability of temperature across and within scenarios in anthropogenic climate change experiments, Climatic Change, 146 (2018), pp. 319–333.
  * [2] A. Basse-O’Connor, S.-E. Graversen, and J. Pedersen, Martingale-type processes indexed by the real line, ALEA Lat. Am. J. Probab. Math. Stat., 7 (2010), pp. 117–137.
  * [3] , Stochastic integration on the real line, Theory Probab. Appl., 58 (2014), pp. 193–215.
  * [4] V. I. Bogachev, Gaussian measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.
  * [5] D. Bolin and K. Kirchner, The rational SPDE approach for Gaussian random fields with general smoothness, J. Comput. Graph. Statist., 29 (2020), pp. 274–285.
  * [6] D. Bolin, K. Kirchner, and M. Kovács, Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise, BIT, 58 (2018), pp. 881–906.
  * [7] , Numerical solution of fractional elliptic stochastic PDEs with spatial white noise, IMA J. Numer. Anal., 40 (2020), pp. 1051–1073.
  * [8] S. G. Cox, M. Hutzenthaler, and A. Jentzen, Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations.  Preprint, arXiv:1309.5595v3, 2021.
  * [9] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, vol. 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, second ed., 2014.
  * [10] J. L. Doob, The elementary Gaussian processes, Ann. Math. Statistics, 15 (1944), pp. 229–282.
  * [11] , Stochastic processes, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990.  Reprint of the 1953 original, A Wiley-Interscience Publication.
  * [12] T. E. Duncan, B. Pasik-Duncan, and B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dyn., 2 (2002), pp. 225–250.
  * [13] K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
  * [14] B. Goldys, S. Peszat, and J. Zabczyk, Gauss-Markov processes on Hilbert spaces, Trans. Amer. Math. Soc., 368 (2016), pp. 89–108.
  * [15] G. R. Grimmett and D. R. Stirzaker, Probability and random processes, Oxford University Press, New York, third ed., 2001.
  * [16] M. Haase, The functional calculus for sectorial operators, vol. 169 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 2006\.
  * [17] L. Herrmann, K. Kirchner, and C. Schwab, Multilevel approximation of Gaussian random fields: fast simulation, Math. Models Methods Appl. Sci., 30 (2020), pp. 181–223.
  * [18] T. Hida and M. Hitsuda, Gaussian processes, vol. 120 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1993.  Translated from the 1976 Japanese original by the authors.
  * [19] D. P. Huy, A remark on non-Markov property of a fractional Brownian motion, Vietnam J. Math., 31 (2003), pp. 237–240.
  * [20] T. Hytönen, J. van Neerven, M. Veraar, and L. Weis, Analysis in Banach spaces. Vol. II, vol. 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer, Cham, 2017.  Probabilistic methods and operator theory.
  * [21] O. Kallenberg, Foundations of modern probability, vol. 99 of Probability Theory and Stochastic Modelling, Springer, Cham, third ed., 2021.
  * [22] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006.
  * [23] K. Kirchner and D. Bolin, Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces, Ann. Statist., 50 (2022), pp. 1038–1065.
  * [24] K. Kirchner and J. Willems, Regularity theory for a new class of fractional parabolic stochastic evolution equations.  Preprint, arXiv:2205.00248v2, 2023.
  * [25] P. C. Kunstmann and L. Weis, Maximal $L_{p}$-regularity for parabolic equations, Fourier multiplier theorems and $H^{\infty}$-functional calculus, in Functional analytic methods for evolution equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, pp. 65–311.
  * [26] A. B. Lawson, Hierarchical modeling in spatial epidemiology, Wiley Interdiscip. Rev. Comput. Stat., 6 (2014), pp. 405–417.
  * [27] F. Lindgren, H. Bakka, D. Bolin, E. Krainski, and H. Rue, A diffusion-based spatio-temporal extension of Gaussian Matérn fields.  Preprint, arXiv:2006.04917v3, 2023.
  * [28] F. Lindgren, D. Bolin, and H. Rue, The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running, Spat. Stat., 50 (2022).  Paper No. 100599.
  * [29] F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol., 73 (2011), pp. 423–498.  With discussion and a reply by the authors.
  * [30] W. Liu and M. Röckner, Stochastic partial differential equations: an introduction, Universitext, Springer, Cham, 2015.
  * [31] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), pp. 422–437.
  * [32] C. Martínez Carracedo and M. Sanz Alix, The theory of fractional powers of operators, vol. 187 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 2001.
  * [33] B. Matérn, Spatial variation: Stochastic models and their application to some problems in forest surveys and other sampling investigations, Meddelanden Från Statens Skogsforskningsinstitut, Band 49, Nr. 5, Stockholm, 1960.
  * [34] A. F. Mejia, Y. Yue, D. Bolin, F. Lindgren, and M. A. Lindquist, A Bayesian general linear modeling approach to cortical surface fMRI data analysis, J. Amer. Statist. Assoc., 115 (2020), pp. 501–520.
  * [35] F. Oberhettinger and L. Badii, Tables of Laplace transforms, Springer-Verlag, New York-Heidelberg, 1973.
  * [36] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.
  * [37] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
  * [38] S. Peszat and J. Zabczyk, Stochastic partial differential equations with Lévy noise, vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007.
  * [39] J. Picard, Representation formulae for the fractional Brownian motion, in Séminaire de Probabilités XLIII, vol. 2006 of Lecture Notes in Math., Springer, Berlin, 2011, pp. 3–70.
  * [40] D. Revuz and M. Yor, Continuous martingales and Brownian motion, vol. 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, third ed., 1999\.
  * [41] Yu. A. Rozanov, Markov random fields, Applications of Mathematics, Springer-Verlag, New York-Berlin, 1982.  Translated from the Russian by Constance M. Elson.
  * [42] H. Rue and L. Held, Gaussian Markov random fields, vol. 104 of Monographs on Statistics and Applied Probability, Chapman & Hall/CRC, Boca Raton, FL, 2005.  Theory and applications.
  * [43] D. Sanz-Alonso and R. Yang, Finite element representations of Gaussian processes: balancing numerical and statistical accuracy, SIAM/ASA J. Uncertain. Quantif., 10 (2022), pp. 1323–1349.
  * [44] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
  * [45] J. van Neerven, Functional analysis, vol. 201 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2022.
  * [46] J. van Neerven and M. C. Veraar, On the stochastic Fubini theorem in infinite dimensions, in Stochastic partial differential equations and applications—VII, vol. 245 of Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 323–336.
  * [47] P. Whittle, Stochastic processes in several dimensions, Bull. Inst. Internat. Statist., 40 (1963), pp. 974–994.


# Quantifying Dynamical Heterogeneity Length Scales of Interface Water across
Model Membrane Phase Transition

Sheeba Malik  Department of Chemistry, IIT Jodhpur, Jodhpur, Rajasthan, India
Smarajit Karmakar  Centre for Interdisciplinary Sciences, Tata Institute of
Fundamental Research, Hyderabad, India  Ananya Debnath  Department of
Chemistry, IIT Jodhpur, Jodhpur, Rajasthan, India<EMAIL_ADDRESS>
###### Abstract

All-atom molecular dynamics simulations of 1,2-dimyristoyl-sn-
glycerol-3-phosphocholine lipid membranes reveal a membrane phase transition
dictated drastic growth in the interface water (IW) heterogeneity length
scales. It acts as an alternate probe to capture the ripple size of the
membrane and follows an activated dynamical scaling with the relaxation time
scale solely within the gel phase. The results quantify the mostly unknown
correlations between the spatio-temporal scales of the IW and membranes at
various phases under physiological and supercooled conditions

††preprint: AIP/123-QED

Lipid bilayers form a continuous semipermeable barrier surrounding cells and
compartmentalize the intracellular space. They are essential for controlling
the dynamics, arrangement, and function of membrane proteins smith2012lipid
at the fluid (or liquid-crystalline, Lα) phase at physiological temperature.
The bilayers at the fluid phase comprise of heterogeneous functional
microdomains that aid in activities including cell signalling, cell adhesion,
and membrane trafficking. Thus, the lateral heterogeneity is accepted as a
criterion for the function of a complex biological membrane, specifically in
the context of a raft jacobsonNature:2007  relevant for signalling cascades
ToomreNature:2000 . Quantifying the heterogeneity in a bio-membrane using the
current biochemical or biophysical tools is an incredibly challenging task due
to the presence of unusual liquid-like domains with microseconds to seconds
time scale and nanometers to micrometers length scales having both short and
long-range orders. Molecular level characterization of such dynamical domains
through experiments is still fragmentary due to the lack of the relevant
atomistic trajectories and the unknown spatio-temporal correlations.
At physiological temperature interface water molecules (IW) near a single
component fluid membrane exhibit signatures of dynamical heterogeneity
DebnathSM:2019  and can probe regional lipid dynamics which is difficult to
access otherwise DebnathPCCP:2020 . This is the stepping stone of the current
study which aims to measure the spatio-temporal scales of different phases of
functionally relevant bilayers through the IW dynamics. Using sum-frequency
generation spectroscopy heterogeneity of IW at the water-charged lipid
interface has been identified Bonn_JPCL_2019 . The fluid or disordered
membrane undergoes a phase transition to a ripple (Pβ) phase which quickly
changes to a gel (Lβ) phase debnathSM:2014  upon cooling. The membrane remains
in the gel phase upon supercooling. Similar disorder to order transition of
the membrane occurs from dehydration induced drastic slow-down in structural
relaxation originated from the dynamical heterogeneity of the interface water
molecules (IW) DebnathJCP:2021 . Understanding cellular membranes at
supercooled temperatures is extremely crucial for cryo-preservation methods,
which have a wide range of applications in food science, pharmacology,
clinical medicine, and the preservation of human embryos for in vitro
fertilization LoisNature:2000 . Despite such innumerable applications,
investigations on the lipid dynamics at the gel phase are rarely found
starrJPCB:2016 . Using QENS, higher contribution of rotational water dynamics
is found near multilamellar DMPC bilayers compared to the translational one at
supercooled temperature swenson2008solvent . Although it is widely accepted
that IW aids in different activities of cellular membranes, very few works
attempt to establish the correlations of hydration dynamics to membrane
functions ChattopadhyayJPCL:2021 . Using coarse-grained (CG) and all-atom (AA)
molecular simulations and different experimental techniques, the phase
behavior of model membranes have been investigated extensively Debnath_JPCB_20
; KlaudaBBA:2018 ; OldfieldBiochemistry:1979; SturtevantPNAS:1976;
HansmaScience:1988; akabori:2015 . However, none of the work mentioned above
focuses on the underlying role of dynamical heterogeneity across phase
transitions. The thermodynamics and dynamics of the IW exhibit sharp cross-
overs across the bilayer melting transition which demonstrates their
correlations debnathPRL:2013 . Thus, the effect of membrane phase transitions
on dynamical heterogeneity of the IW is fundamentally important and long
overdue. Although many theoretical models Sarika_PNAS_08  and numerical
methods are developed to study glass transition phenomena, their
applicabilities on the membrane phase transitions and water dynamics are not
understood. The current research finds out the relevance of different theories
Thirumalai_PRL_1987  and recent numerical approaches karmakarPNAS:2009  of
glass dynamics on model membranes across phase transitions. The objective is
many fold: a) to gain a fundamental understanding of the dynamical
heterogeneity of the IW and the lipid membranes near phase transitions, b) to
establish relations of the spatio-temporal scales of the IW to that of the
membranes, c) to build up an alternate method through IW dynamics to probe
membrane domains under physiological and stressed conditions. Furthermore,
this strategy will help in comprehending the domain length scale of a raft,
which is rarely found in theory and extremely difficult to gain from an
experiment.
Therefore, $11.55~{}\mu$s long all-atom molecular dynamics (MD) simulations of
dimyristoylphosphatidylcholine (DMPC) bilayers are performed at various
temperatures and two different system sizes ( $6\times 6$ nm2 and $18\times
18$ nm2) to cover the three primary phases. Simulation details are mentioned
in section I of the Supporting Information (SI) and the runlengths of all
bilayers are mentioned in table S1 of the SI. CHARMM36 force fields are used
for the 128 DMPC lipids solvated in $5743$ TIP4P/2005 JLF:2005 ($6\times 6$
nm2 box size) water molecules. This particular pairing of the lipid force-
fields and the water model accurately reproduces the fluid to ripple to gel
phase transitions klauda2010update . The area per head groups ($a_{h}$) of
membranes from our simulations at all phases match well with the available
literature KlaudaBBA:2014 ; KlaudaBBA:2018 ; KatsarasBBA:2011 (section II and
table S2 of the SI). Furthermore, the governing mechanism of dynamical
heterogeneity of supercooled water is unaffected by the choice of water model
KimScienceAdvances:2017 ; galamba2016hydrogen .
Snapshots of the equilibrated bilayers at three phases are shown in figure 1
(a-c) and the bilayers at the remaining temperatures are shown in figure S1 of
the SI (considered for the calculations of structural relaxations). The
$a_{h}$ drops abruptly from $308$ K to $292$ K and from $284$ K to $268$ K
demonstrating the fluid (Lα) to the ripple (Pβ) to the gel ($L_{\alpha}$)
phase transition temperatures (figure S2 of the SI). The ripple phase is
identified from the existence of periodic interdigitated and non-
interdigitated regions of lipid tails 2005molecular. The homogeneous thickness
and interdigitation of the fluid phase ($324$ K-$303$ K) (figure 1 (d) and
(g)) become periodically heterogeneous for the ripple phase ($292$ K-$284$ K)
((e) and (h)) which disappears for the gel phase ($273$ K-$218$ K). The region
with smaller thickness is correlated to the higher interdigitation at all
phases. The thickness and interdigitation of the remaining temperatures are
shown in figure S4 in the SI (interdigitation). At the ripple phase, there is
a coexistence of highly interdigitated ordered domain followed by a fully
expanded disordered domain (shown in S4 in the SI). The ripple wavelength of
$\sim 4.2$ nm from figure 1 (e and h) matches well with that obtained from the
spectral intensity of thickness and interdigitation fluctuations discussed in
section II of the SI.

Figure 1: Snapshots of bilayers at three phases (a-c), thickness (d-f) and
interdigitation (g-i) superimposed on voronoi area per lipid head
respectively. Color code with representation: DMPC: CPK representation in blue
and green for two leaflets; IW water: VDW (van der Waals) representation in
red. Figure 2: Self intermediate scattering function ($F_{s}(q,t)$) of a) BW,
b) IW and c) P heads respectively at $\lambda$ = $0.5nm$

The homogeneous thickness and interdigitation of the fluid phase ($324$
K-$303$ K) (figure 1 (d) and (g)) become periodically heterogeneous for the
ripple phase ($292$ K-$284$ K) ((e) and (h)) which disappears for the gel
phase ($273$ K-$218$ K). The region with smaller thickness is correlated to
the higher interdigitation at all phases. The thickness and interdigitation of
the remaining temperatures are shown in figure S4 in the SI (interdigitation).
At the ripple phase, there is a coexistence of highly interdigitated ordered
domain followed by a fully expanded disordered domain (shown in S4 in the SI).
The ripple wavelength of $\sim 4.2$ nm from figure 1 (e and h) matches well
with that obtained from the spectral intensity of thickness and
interdigitation fluctuations discussed in section II of the SI. As the largest
survival time of the water molecule near membranes at the highest temperature
is found to be $\sim 100$ ps DebnathCPL:2022 , it is considered as the minimum
residence time of the IW at all temperatures. Thus, if a water molecule is
present within $\pm 0.3$ nm of the lipid P head atoms continuously for $100$
ps, it is classified as an IW (section III of the SI). To find out the
structural relaxation, self intermediate scattering functions (SISF) of the
BW, IW and lipid heads are calculated (section IV of the SI). The initial
decay of the SISF (figure 2 a)-c)) is due to the ballistic or cage motion
followed by the emergence of Boson peaks at lower temperatures ($228-218$ K),
as observed commonly in supercooled liquids. The time at which
$F_{s}(q,t)=1/e$, is referred to as the $\alpha$ relaxation time-scale,
$\tau_{\alpha}$. With lowering the temperature, $\tau_{\alpha}$ start growing
for the BW, the IW and the P heads. The growths are significant for the IW and
the P once the temperature goes below $273$ K and the membrane is at the gel
phase. The growing $\tau_{\alpha}$ of the IW and the P heads are strongly
correlated at all three phases of the membranes even at super-cooled
temperatures (figure S5 in the SI). Block analysis has been employed on the
membrane interface with $18\times 18$ nm2 surface area (Table S1 of the SI).
The non-Gaussian van Hove distributions, $G_{s}(X,t)$, of the IW on the
membrane surface become more and more Gaussian as the $L_{B}$ increases (see
figure 3 a), d), and g) for the fluid, ripple, and gel phases, respectively
and figure S6 of the SI for remaining temperatures). Binder cumulant,
$B(L{{}_{B}},T)=0$ for higher temperatures and larger blocks as the underlying
distribution is Gaussian. With decreasing block size, the values of the
$B(L{{}_{B}},T)$ increase at all temperatures due to larger deviations from
Gaussianity seen in figure 3 b), e), and h) for three phases of the membrane.
As the binder cumulant is a scaling function of the underlying correlation
length, it is assumed to follow the relation of
$B(L{{}_{B}},T)=f[L_{B}/\xi(T)]$. Thus, $B(L{{}_{B}},T)$ at different
temperatures are subjected to a data collapse and plotted in figure 3 c), f)
and i) for the fluid, ripple, and gel phases, respectively. The insets of
figure 3 depict the temperature dependence of the scaling length scale, $\xi$,
known as heterogeneity length scale. For the fluid Lα phase, $\xi$ increases
with a decrease in $T$. The dependence disappears for the ripple Pβ phase
unlike the glassy liquids.

Figure 3: van Hove function, $G_{s}(X,t)$, of the IW for block lengths of
$9-1$ nm for a) 324 K, d) 284 K and g) 218 K at the fluid, ripple and gel
phases respectively. Binder cumulant of the blocked van Hove function at
previously mentioned three temperatures in b), e) and h) respectively. Finite-
size scaling of the binder cumulant by data collapse at three temperatures c),
f), and i). Respective insets show the temperature dependence of the
heterogeneity length scale, $\xi$, obtained from the data collapse of the
binder cumulant at three phases. Figure 4: $g_{1}$ versus $\xi$ for a) Lα, b)
Pβ, and c) Lβ phase. $g_{1}$ follows power law with $\xi$ where the exponents
are $2.41$, $3.66$, and $13.24$ for three respective phases. d) Temperature
dependence of $\xi$ of the IW showing sudden growth in $\xi$ at the phase
transition temperatures. e) Dependence of log $\tau^{*}$ on
$\epsilon\xi/k_{B}T$ follows an activated dynamic scaling for the Lβ phase.
Here, $\tau^{*}=\tau_{\alpha}\sigma^{-1}\sqrt{\epsilon/M}$. $\sigma,\epsilon$,
and $M$ of oxygens of water molecules are considered for the calculation.

The values of $\xi$ at the ripple phase are close to the half of the ripple
wavelengths as obtained from the spectra of thickness or interdigitation at
all temperatures (figure S7 of the SI). This signifies that the periodic patch
or domain of the ripple of the membrane can be captured by the heterogeneity
length scale of the IW which do not vary with temperature within the same
phase. The $T$ dependency of the $\xi$ is very weak at the gel phase. The
heterogeneity length scale follows a power law behavior with the amplitude of
the first peak of the radial distribution functions, $g_{1}$ DebnathCPL:2022
of the IW at each of the three phases of membranes (figure 4 a), b) and c))
suggesting that the correlation length scale is structure dominated. Figure 4
d) shows that the change in $\xi$ is minor within a given phase but the length
scale significantly grows once there is a phase transition. However, within a
given phase, there is a drastic slow-down in the $\tau_{\alpha}$ with
decreasing temperature without a significant change in $\xi$ (figure 4 e). On
the contrary, there are sharp changes in $\xi$, across the membrane phase
transitions. Thus the heterogeneity time scales are slowed down mainly by
supercooling which is not accompanied by a similar growth in the heterogeneity
length scales. The large growth in the length scales is governed by the
membrane phase transitions. The relaxation time scale, $\tau_{\alpha}$ of the
IW obeys the relation, $\tau_{\alpha}\sim e^{a\xi/T},~{}a=$constant, solely
near the gel phase (figure 4 f)) which are not valid at the fluid or the
ripple phase (figure S8 of the SI).

In conclusion, the study uses $11.55\mu$s all-atom molecular dynamics
simulations to correlate and quantify the spatio-temporal heterogeneities of
the IW to the DMPC lipid membranes upon fluid-to-ripple-to-gel phase
transitions. Although there is a $\alpha$-relaxation time lag between the IW
and the lipids, the lipid heads and the IW are strongly correlated to each
other at all three phases with a strong enhancement in dynamical heterogeneity
upon supercooling. The length scales of spatially heterogeneous dynamics of
the IW are quantified using a block analysis technique, and then correlated to
the lateral organizations of the membranes across phase transitions. One-
dimensional van Hove correlation functions of the IW shows a shift from the
non-Gaussianity to the Gaussianity once the block size increases at each phase
of the membrane. The deviations from the Gaussianity are estimated by Binder
cumulant which increases with lower block size at all temperatures.
Remarkably, the size of the rippling of the membrane can be obtained from the
heterogeneity length scale of the IW. The length scale does not change
significantly upon supercooling at the gel phase and follows an activated
dynamical scaling with the relaxation time scale. This observation is in
similar line to what is predicted by the random-field Ising model Young_98 .
The drastic growth in the $\xi$ is dominated by the membrane phase transitions
and the drastic slow-down in the relaxation time scale is dictated by the
supercooling. Thus, the dynamic cross-over in the heterogeneity length scale
of the IW will provide a new avenue to identify the micro-domain which can be
critical for membrane functionalities. Since probing nano-domains in
biological membranes are extremely challenging and no current biophysical tool
is available yet to this end jacobsonNature:2007 , our method opens up a new
direction to characterize functionally relevant nano-domains of membranes
through the heterogeneity length scale of the IW. As per our knowledge, for
the first time, a systematic analysis is performed to estimate the spatio-
temporal correlations of the hydration water to the model lipid membranes
across phases relevant to physiological and low-temperature stressed
conditions. The findings can help us to better understand bio-protection
processes under extreme conditions and will aid future research on domain-
associated transport and signaling in biomembranes at low temperatures.

See the supplementary material for the simulation methods including tables,
and texts explaining the identification of the interface water, survival
probability, bilayer properties, and figures. The supplementary files are
available at

A.D. acknowledges the financial support of the grant SERB CRG/2019/000106. S.M
is thankful to Dr. Indrajit Tah for useful discussions and to Prof. Jeffery B.
Klauda for providing the pdb of the equilibrated DMPC bilayer (with 72 lipids)
at $273$ K.

AUTHOR DECLARATIONS
Conflict of Interest
The authors declare no competing interest.
Author Contributions
ddS.M. performed the molecular dynamics simulations, analysis and wrote the
original manuscript. S.K. supervised the project and manuscript editing. A.D.
conceived the project, manuscript writing, project supervision, and funding
acquisition.
DATA AVAILABILITY
The data that support the findings of this study are available within the
article and its supplementary material.

## References

  * [1] Adam W Smith.  Lipid–protein interactions in biological membranes: a dynamic perspective.  Biochim. Biophys. Acta - Biomembr., 1818(2):172–177, 2012.
  * [2] Ken Jacobson, Ole G Mouritsen, and Richard GW Anderson.  Lipid rafts: at a crossroad between cell biology and physics.  Nat. Cell Biol., 9(1):7–14, 2007.
  * [3] Kai Simons and Derek Toomre.  Lipid rafts and signal transduction.  Nat. Rev. Mol. Cell Biol., 1(1):31–39, 2000.
  * [4] Abhinav Srivastava, Smarajit Karmakar, and Ananya Debnath.  Quantification of spatio-temporal scales of dynamical heterogeneity of water near lipid membranes above supercooling.  Soft Matter, 15(47):9805–9815, 2019.
  * [5] Abhinav Srivastava, Sheeba Malik, Smarajit Karmakar, and Ananya Debnath.  Dynamic coupling of a hydration layer to a fluid phospholipid membrane: intermittency and multiple time-scale relaxations.  Phys. Chem. Chem. Phys., 22(37):21158–21168, 2020.
  * [6] Takakazu Seki, Shumei Sun, Kai Zhong, Chun-Chieh Yu, Kevin Machel, Lisa B. Dreier, Ellen H. G. Backus, Mischa Bonn, and Yuki Nagata.  Unveiling heterogeneity of interfacial water through the water bending mode.  The Journal of Physical Chemistry Letters, 10(21):6936–6941, 2019\.
  * [7] Ananya Debnath, Foram M Thakkar, Prabal K Maiti, V Kumaran, and KG Ayappa.  Laterally structured ripple and square phases with one and two dimensional thickness modulations in a model bilayer system.  Soft Matter, 10(38):7630–7637, 2014.
  * [8] Sheeba Malik and Ananya Debnath.  Dehydration induced dynamical heterogeneity and ordering mechanism of lipid bilayer.  J. Chem. Phys., 154(17):174904, 2021.
  * [9] John H Crowe and Lois M Crowe.  Preservation of mammalian cells—learning nature’s tricks.  Nat. Biotechnol, 18:145–146, 2000.
  * [10] Neha Shafique, Kiley E Kennedy, Jack F Douglas, and Francis W Starr.  Quantifying the heterogeneous dynamics of a simulated dipalmitoylphosphatidylcholine (dppc) membrane.  J. Phys. Chem. B, 120(23):5172–5182, 2016.
  * [11] Jan Swenson, Florian Kargl, Peter Berntsen, and Christer Svanberg.  Solvent and lipid dynamics of hydrated lipid bilayers by incoherent quasielastic neutron scattering.  J. Chem. Phys., 129(4):07B616, 2008.
  * [12] Sreetama Pal and Amitabha Chattopadhyay.  Hydration dynamics in biological membranes: emerging applications of terahertz spectroscopy.  J. Phys. Chem. Lett., 12(39):9697–9709, 2021.
  * [13] Arpita Srivastava and Ananya Debnath.  Asymmetry and rippling in mixed surfactant bilayers from all-atom and coarse-grained simulations: Interdigitation and per chain entropy.  J. Phys. Chem. B, 124(29):6420–6436, 2020.
  * [14] Pouyan Khakbaz and Jeffery B Klauda.  Investigation of phase transitions of saturated phosphocholine lipid bilayers via molecular dynamics simulations.  Biochim. Biophys. Acta - Biomembr., 1860(8):1489–1501, 2018.
  * [15] Kiyotaka Akabori and John F Nagle.  Structure of the dmpc lipid bilayer ripple phase.  Soft Matter, 11(5):918–926, 2015.
  * [16] Ananya Debnath, KG Ayappa, and Prabal K Maiti.  Simulation of influence of bilayer melting on dynamics and thermodynamics of interfacial water.  Phys. Rev. Lett., 110(1):018303, 2013.
  * [17] Sarika Maitra Bhattacharyya, Biman Bagchi, and Peter G. Wolynes.  Facilitation, complexity growth, mode coupling, and activated dynamics in supercooled liquids.  Proc. Natl. Acad. Sci., 105(42):16077–16082, 2008.
  * [18] T. R. Kirkpatrick and D. Thirumalai.  Dynamics of the structural glass transition and the p-spin—interaction spin-glass model.  Phys. Rev. Lett., (20):2091, 1987.
  * [19] Smarajit Karmakar, Chandan Dasgupta, and Srikanth Sastry.  Growing length and time scales in glass-forming liquids.  Proc. Natl. Acad. Sci., 106(10):3675–3679, 2009.
  * [20] J. L. F. Abascal and C. Vega.  A general purpose model for the condensed phases of water: Tip4p/2005.  J. Chem. Phys., 123:234505–1–12, 2005.
  * [21] Jeffery B Klauda, Richard M Venable, J Alfredo Freites, Joseph W O’Connor, Douglas J Tobias, Carlos Mondragon-Ramirez, Igor Vorobyov, Alexander D MacKerell Jr, and Richard W Pastor.  Update of the charmm all-atom additive force field for lipids: validation on six lipid types.  J. Phys. Chem. B, 114(23):7830–7843, 2010.
  * [22] Xiaohong Zhuang, Judah R Makover, Wonpil Im, and Jeffery B Klauda.  A systematic molecular dynamics simulation study of temperature dependent bilayer structural properties.  Biochim. Biophys. Acta - Biomembr., 1838(10):2520–2529, 2014.
  * [23] Norbert Kučerka, Mu-Ping Nieh, and John Katsaras.  Fluid phase lipid areas and bilayer thicknesses of commonly used phosphatidylcholines as a function of temperature.  Biochim. Biophys. Acta - Biomembr., 1808(11):2761–2771, 2011.
  * [24] Takeshi Kawasaki and Kang Kim.  Identifying time scales for violation/preservation of stokes-einstein relation in supercooled water.  Sci. Adv., 3(8):e1700399, 2017.
  * [25] N Galamba.  On the hydrogen-bond network and the non-arrhenius transport properties of water.  Journal of Physics: Condensed Matter, 29(1):015101, 2016.
  * [26] Sheeba Malik and Ananya Debnath.  Structural changes of interfacial water upon fluid-ripple-gel phase transitions of bilayers.  Chem. Phys. Lett., 799:139613, 2022.
  * [27] A. P. Young.  Spin glasses and random fields.  12, 1998.


# A primality test for $Kp^{n}+1$ numbers and a generalization of Safe Primes
and Sophie Germain Primes

A. Ramzy  Department of Mathematics, Faculty of Education, Azhar University,
Cairo, Egypt<EMAIL_ADDRESS>
###### Abstract.

In this paper, we provide a generalization of Proth’s theorem for integers of
the form $Kp^{n}+1$. In particular, a primality test that requires only one
modular exponentiation similar to that of Fermat’s test without the
computation of any GCD’s. We also provide two tests to increase the chances of
proving the primality of $Kp^{n}+1$ numbers (if they are primes indeed). As a
corollaries of these tests we provide three families of integers $N$ whose
primality can be certified only by proving that $a^{N-1}\equiv 1\pmod{N}$
(Fermat’s test). We also generalize Safe Primes and define those generalized
numbers as $a$-SafePrimes for being similar to SafePrimes (since $N-1$ for
these numbers has large prime factor the same as SafePrimes), we address some
questions regarding the distribution of those numbers and provide a conjecture
about the distribution of their related numbers $a$-SophieGermainPrimes which
seems to be true even if we are dealing with $100$, $1000$, or $10000$ digits
primes.

###### Key words and phrases:

Primality test, Safe prime, Sophie Germain prime

###### 1991 Mathematics Subject Classification:

Primary 11Y11, 11N80, 11N05

##  1\. INTRODUCTION

One of the fundamental facts about prime numbers is Fermat’s result that if
$N$ is prime then for every integer $a$ we have

(1.1) $a^{N}\equiv a\pmod{N}.$

This leads to the well-known Fermat’s primality test which is a probabilistic
test, but since the verification of (1.1) for a given $a$ and $N$ is
computationally inexpensive, even for large $N$, there has been a number of
results studying additional conditions that could be combined with Fermat’s
test to conclude primality. For example, in 1878 Proth (see [6]) presented the
following theorem to determine whether $N$ is prime or not when $N$ is of the
form $N=K2^{n}+1$ (Proth numbers)

######  Theorem 1.1 (Proth, 1878).

Let $N=K2^{n}+1$, where $K$ is odd and $K<2^{n}$. If there exists an integer
$a>1$, such that $a^{\frac{N-1}{2}}\equiv-1\pmod{N}$, then $N$ is prime.

In 1914 an important and useful result was proved by Pocklington, which only
requires a partial factorization of $(N-1)$

######  Theorem 1.2 (Pocklington, 1914).

Let $N-1=q^{n}R$ where $q$ is a prime, $n\geq 1$, and $q\nmid R$. Assume that
there exists an integer $a>1$ such that:

  1. (i)

$a^{N-1}\equiv 1\pmod{N}$, and

  2. (ii)

gcd$(a^{\frac{N-1}{q}}-1,N)=1$.

Then each prime factor of $N$ is of the form $mq^{n}+1$, with $m\geq 1$.

As a result of 1.2 he gave the first generalization of Proth’s theorem
suitable for numbers of the form $N=Kp^{n}+1$, which are called _Generalized
Proth numbers_ (see [6]):

######  Theorem 1.3 (Pocklington, 1914).

Let $N=Kp^{n}+1$ with $K<p^{n}$, and $p$ is prime. If there exists an integer
$a>1$ such that:

  1. (i)

$a^{N-1}\equiv 1\pmod{N}$, and

  2. (ii)

gcd$(a^{\frac{N-1}{p}}-1,N)=1$,

then $N$ is prime.

In [5] the following simpler generalization was presented by Grau, Oller-
Marcen, and Sadornil.

######  Theorem 1.4 (Grau et al., 2015).

Let $N=Kp^{n}+1$, where $p$ is prime and $K<p^{n}$. Assume that
$a\in\mathbb{Z}$ is a $p$-th power non-residue, then $N$ is a prime if and
only if $\Phi_{p}(a^{\frac{N-1}{p}})\equiv 0\pmod{N}$.

Then they gave a more useful test which increases the chance of proving the
primality of $N$ (if $N$ is prime indeed). Note that we replaced $(J)$ in the
original theorem by $(n-j)$ in order to compare it to theorem 2.5 in the
following section, namely, the conditions $n-1\geq j\geq 0$ and
$2(n-j)>\log_{p}(K)+n$ in the following theorem are equivalent to the
conditions $1\leq J\leq n$ and $2J>\log_{p}(K)+n$ in the original theorem in
[5].

######  Theorem 1.5 (Grau et al., 2015).

Let $N=Kp^{n}+1$, where $p$ is prime. If there exists $n-1\geq j\geq 0$ such
that:

  * (i)

$\Phi_{p}(a^{Kp^{n-j-1}})\equiv 0\pmod{N}$

  * (ii)

$2(n-j)>\log_{p}(K)+n$.

Then $N$ is prime.

Theorem 1.4 states that if $\Phi_{p}(a^{\frac{N-1}{p}})\equiv 0\pmod{N}$ such
that $N=Kp^{n}+1$, then $N$ is prime. In practice, $(\Phi_{p}(x))$ is easily
computed as $\frac{x^{p}-1}{x-1}$. Thus, to use Theorem 1.4 we would need to
verify that $a^{N-1}\equiv 1\pmod{N}$ and that $a^{\frac{N-1}{p}}-1$ is
invertible $\pmod{N}$. The latter condition can be verified by checking that
gcd$(a^{\frac{N-1}{p}}-1,N)=1$, which is well known to be an $O(\log N)$
computation, the same can be said as for computing $\Phi_{p}(a^{Kp^{n-j-1}})$
in 1.5 since it requires the gcd step. Note that we can compute $\Phi_{p}(x)$
as $1+x+x^{2}+\cdots+x^{p-1}$ but only when $p$ is small, since computing
$\Phi_{p}(x)$ that way would be time consuming if $p$ is large.

We organized the paper as follows. In Section 2 we prove a couple of simple
lemmas that we use to prove (Theorem 2.3) as well as a useful generalization
of it (Theorem 2.5), we also put a Remark at the end of the Section to clarify
the main contributions of Theorems 2.3 and 2.5 and provide a simple
comparisons to existing methods. In Section 3 we provide two more useful tests
but only for integers of the form $Kp^{n}+1$ such that $p^{j}\geq
p^{n-j}\geq(N-1)^{1/3}$ or $p^{j}\geq p^{2(n-j)}$ and
$p^{n-j}\geq(N-1)^{2/7}$, and after each theorem we put a simple comparisons
to existing methods, note that the above conditions on $p^{n-j}$ will be
satisfied mostly, since in most cases we will have $p^{n}$ be much larger than
$K$. Also as a corollaries we give two infinite families of integers $N$ whose
primality can be certified only by proving that $a^{N-1}\equiv 1\pmod{N}$. In
section 4 we give a family of integers similar to Safe Primes (since $N-1$ for
these numbers has large prime factor the same as Safe primes), those numbers
will be defined as $a$-SafePrimes, we address some questions regarding the
distribution of those classes of primes and provide some computations for
$2$-SafePrimes, and the distribution of the related numbers
$2$-SophieGermainPrimes will be addressed in section 5. In Section 5 we
provide some estimates to find the probability that a given prime is
$a$-SophieGermainPrime, then we give some computations for
$2$-SophieGermainPrimes which shows that the accuracy of the estimates is very
acceptable, even if we are dealing with $100$, $1000$ or $10000$ digits primes
(random or consecutive), we also give a conjecture about the distribution of
$2$-SophieGermainPrimes (and $a$-SophieGermainPrimes in general) which helped
us to discover many primes larger than $10^{999}$ and some of them were larger
than $10^{9999}$.

##  2\. A Generalization of Proth’s Theorem

In this section we shall state and prove theorems 2.3 and 2.5, whose provide a
simple primality test for generalized Proth’s numbers $N=Kp^{n}+1$. To prove
the theorems we require two lemmas.

######  Lemma 2.1.

Assume that $A,P$ are integers with $1\leq A\leq P$. If there is an integer
$D>0$ such that

$\frac{AP+1}{DP+1}\in\mathbb{Z},$

then we must have $D=A$.

###### Proof.

If $DP+1\mid AP+1$ we must have $D\leq A$. Write $A=cD+r$ with $c>0$ and
$0\leq r<D$. Hence

$\displaystyle\frac{(cD+r)P+1}{DP+1}$
$\displaystyle=\frac{cDP+rP+1+c-c}{DP+1}$
$\displaystyle=c+\frac{rP+1-c}{DP+1}.$

Note that $c\leq A\leq P$, so if $r\geq 1$ we will have $rP-c+1\geq 1$, hence
the numerator of the last fraction is a positive number which is strictly less
than its denominator (since $r<D$). Thus the fraction can only be an integer
when $r=0$ and $c=1$, proving the result. ∎

Recall that the _order_ of $a\pmod{N}$ is the least positive integer $m$ such
that $a^{m}\equiv 1\pmod{N}$. We shall denote the order of $a\pmod{N}$ with
$\operatorname{ord}_{N}(a)$.

######  Lemma 2.2.

Assume that $p^{n}\mid\operatorname{ord}_{N}(a)$ where $p$ is prime and
gcd$(p,N)=1$, then there must exist a prime divisor $q$ of $N$ such that
$p^{n}\mid\operatorname{ord}_{q}(a)$, therefore $p^{n}\mid q-1$ .

###### Proof.

Assume that the prime factorization of $N=q_{1}^{e_{1}}\cdots q_{s}^{e_{s}}$,
then
$\operatorname{ord}_{N}(a)=\operatorname{lcm}(\operatorname{ord}_{q_{1}^{e_{1}}}(a),\cdots,\operatorname{ord}_{q_{s}^{e_{s}}}(a))$,
but since $p^{n}\mid\operatorname{ord}_{N}(a)$, then there must exist a prime
divisor $q_{c}$ of $N$ such that
$p^{n}\mid\operatorname{ord}_{q_{c}^{e_{c}}}(a)$, but since
$\operatorname{ord}_{q_{c}^{e_{c}}}(a)=q_{c}^{k}\times\operatorname{ord}_{q_{c}}(a)$,
where $k\leq e_{c}-1$, and since gcd$(p,N)=1$, then $p^{n}$ must divide
$\operatorname{ord}_{q_{c}}(a)$, therefore, $p^{n}\mid q_{c}-1$ . ∎

######  Theorem 2.3.

Let $N=Kp^{n}+1$, where $p$ is prime, $p^{n}\geq K$. Assume that there exists
an integer $a$ such that:

  1. (i)

$a^{Kp^{n-1}}\equiv L\neq 1\pmod{N}$, and

  2. (ii)

$L^{p}\equiv 1\pmod{N}$.

Then $N$ is prime.

###### Proof.

Assume both of (i) and (ii) hold, then $p^{n}\mid\operatorname{ord}_{N}(a)$,
but since gcd$(p,N)=1$, then there must exist a prime divisor $q$ of $N$ such
that $p^{n}\mid q-1$, due to Lemma 2.2, hence $q=hp^{n}+1$ divides $N$
implying $\frac{Kp^{n}+1}{hp^{n}+1}\in\mathbb{Z}$, but due to Lemma 2.1 we
must have $h=K$, therefore, $q=N$ and $N$ is prime. ∎

As corollary, we show that the primality of $2Kp+1$ numbers with
$2K<\log_{a}(2Kp+1)$ can be certified only by proving that $a^{N-1}\equiv
1\pmod{N}$, and then we generalize 2.3.

######  Corollary 2.4.

Let $N=2Kp+1$, where $p$ is prime, $2K\leq p$, and $2K<\log_{a}(2Kp+1)$. Then
$N$ is prime if and only if

(2.1) $a^{N-1}\equiv 1\pmod{N}$

###### Proof.

By the hypothesis we have $2K<\log_{a}(2Kp+1)$, hence $a^{2K}\equiv L\neq
1\pmod{N}$, and $a^{N-1}\equiv 1\pmod{N}$. Thus all the conditions of Theorem
2.3 are satisfied and $N$ is prime. The converse is simply Fermat’s primality
test. ∎

######  Theorem 2.5.

Let $N=Kp^{n}+1$, where $p$ is prime, $p^{n-j}\geq Kp^{j}\Longrightarrow
p^{2(n-j)}\geq Kp^{n}\Longrightarrow 2(n-j)\geq\log_{p}(K)+n$ where $(0\leq
j\leq n-1)$. Assume that there exists an integer $a>1$ such that:

  1. (i)

$a^{Kp^{n-j-1}}\equiv L\neq 1\pmod{N}$, and

  2. (ii)

$L^{p^{j+1}}\equiv 1\pmod{N}$.

Then $N$ is prime.

###### Proof.

As in the proof of Theorem 2.3 we can deduce that there must exist a prime
divisor $q$ of $N$ such that $p^{n-j}\mid q-1$, hence $q=hp^{n-j}+1$ divides
$N$, which means that $\frac{Kp^{j}.p^{n-j}+1}{hp^{n-j}+1}\in\mathbb{Z}$, but
due to Lemma 2.1 we must have $h=Kp^{j}$, therefore, $q=N$ and $N$ is prime. ∎

######  Remark 2.6.

The main contribution of Theorem 2.3 is to show that the gcd step in 1.4 can
in fact be replaced by simply verifying that $a^{\frac{N-1}{p}}\neq
1\pmod{N}$. Computationally and theoretically this provides a simpler
generalization of Proth’s theorem. Note that for small values of $p$ we can
compute $\Phi_{p}(X)$ as $1+X+X^{2}+\cdots+X^{p-1}$ and then theorem 1.4 will
not require any gcd computations and will be equivalent to 2.3
(computationally).

As an example to illustrate Theorem 2.3, consider $N=2\cdot
107^{3}+1=2450087$. We can verify that (i) $2^{2\cdot 107^{2}}\equiv
1302367\pmod{2450087}$ and (ii)$1302367^{107}\equiv 1\pmod{2450087}$, and thus
the hypothesis of Theorem 2.3 are satisfied and we deduce that $N$ is prime.
The same can of course be obtained from Theorem 1.4 by computing
$\Phi_{107}(2^{2\cdot 107^{2}})\pmod{2450087}$, but the latter requires the
additional step of verifying that gcd$(1302367-1,2450087)=1$ (since $p=107$ is
relatively big) in order have a meaningful “denominator" when evaluating
$\phi_{107}$ $\pmod{2450087}$.

The same can be said as for the comparison of 2.5 to 1.5, since if $p$ is big,
then 1.5 will require the gcd step, since then it will be difficult to compute
$\Phi_{p}(a^{Kp^{n-j-1}})$ as
$1+(a^{Kp^{n-j-1}})+(a^{Kp^{n-j-1}})^{2}+\cdots+(a^{Kp^{n-j-1}})^{p-1}$. Note
that if $N$ is indeed prime, then the chances of proving the primality of $N$
are even whether we used 1.5 or 2.5, since if $N=Kp^{n}+1$ then the base ($a$)
will fail to satisfy condition (i) in theorems 2.5 and 1.5 only when ($a$)
itself is a $p^{j+1}$-th power residue modulo $N$, which happens exactly for
$\frac{1}{p^{j+1}}$ of the possible choices for ($a$), which means that
condition (i) in 2.5 and 1.5 will be satisfied by a number of bases that
equals to $Kp^{n}-Kp^{n-j-1}$, but since ($j$) in 2.5 can be as large as ($j$)
in 1.5 then the number of the satisfying bases of (i) in both of 2.5 and 1.5
is the same.

##  3\. Increasing the chances of proving the primality of $Kp^{n}+1$ primes

In this section we provide two tests for $Kp^{n}+1$ numbers such that
$p^{j}\geq p^{n-j}\geq(N-1)^{1/3}$ or $p^{j}\geq p^{2(n-j)}$ and
$p^{n-j}\geq(N-1)^{2/7}$, those two tests can increase the chances of proving
the primality of $Kp^{n}+1$ primes by increasing the number of the satisfying
bases ($a$) of their conditions of primality. We shall give an example after
each theorem to show their main contributions.

######  Theorem 3.1.

Let $N=Kp^{n}+1\neq p^{3(n-j)}+1$, where $p$ is prime and $p^{j}\geq
p^{n-j}\geq(N-1)^{1/3}$. If there exists an integer $a>1$ such that:

  1. (i)

$a^{Kp^{n-j-1}}\equiv L\neq 1\pmod{N}$, and

  2. (ii)

$L^{p^{j+1}}\equiv 1\pmod{N}$.

Then $N$ is prime.

###### Proof.

Assume both of (i) and (ii) hold, then $p^{n-j}\mid\operatorname{ord}_{N}(a)$,
but since $gcd(p,N)=1$, then there must exist a prime divisor $q$ of $N$ such
that $p^{n-j}\mid q-1$, due to Lemma 2.2, therefore, we have two cases:

  1. (i)

$q=N$ and $N$ is prime, or

  2. (ii)

$q<N$ and $N$ is composite.

Now, we will try to exclude (ii) by contradiction. In(ii) we have $q\mid N$,
hence, $\frac{N}{q}=M\in\mathbb{Z}$, but since $q=ap^{n-j}+1$ then we must
have $M=bp^{n-j}+1$ as well, now, if we multiply $q$ by $M$ we will get
$N=Kp^{n}+1=abp^{2(n-j)}+(a+b)p^{n-j}+1\Longrightarrow Kp^{j}=abp^{n-j}+a+b$,
therefore we must have $p^{n-j}\mid(a+b)\Longrightarrow a+b\geq
p^{n-j}\Longrightarrow a+b\geq(N-1)^{1/3}$, but since $ab$ can’t exceed or
equal to $(N-1)^{1/3}$, and since $ab\geq a+b-1$, then the only chance we can
have $ab<(N-1)^{1/3}$ and $p^{n-j}\mid(a+b)$ is when $p^{n-j}=(N-1)^{1/3}$,
$ab=p^{n-j}-1$ and $a+b=p^{n-j}\Longrightarrow N=p^{3(n-j)}+1$, but since we
excluded the latter case by the assumption on $N$, then we must have $q=N$ and
$N$ is prime.

∎

To see the utility of theorem 3.1 you should recall that if $N=Kp^{n}+1$ is
prime, then $a$ will fail to satisfy condition (i) of Theorems 1.5 2.5 and 3.1
only when $a$ itself is a $p^{j+1}$-th power residue modulo $N$, which happens
exactly for $\frac{1}{p^{j+1}}$ of the possible choices for $a$, but $j$ in
3.1 can be larger than $j$ in theorems 2.5 and 1.5. Hence (i) in 3.1 is
satisfied by more bases, which equals to $Kp^{n}-Kp^{n-j-1}$. To give a
concrete example, consider $N=2\cdot 3^{17}+1$. Then according to theorems 1.5
and 2.5 we can only have $j\leq 8$, however taking $1\leq j\leq 8$ and
$a=136837116$ will not satisfy (i) in neither 1.5 nor 2.5. But in 3.1 we can
have $j\leq 11$, and by taking $j=9$ for example, we obtain $136837116^{2\cdot
3^{7}}\equiv 216758952\pmod{N}$ and the result follows from Theorem 3.1.

As a corollary, we give the following class of integers $N$ whose primality
can be certified only by proving that $a^{N-1}\equiv 1\pmod{N}$ (Fermat’s
test).

######  Corollary 3.2.

Let $N=2Kp^{2}+1$, where $p$ is prime, $2K<p$ and $2K<\log_{a}(2Kp^{2}+1)$ .
If there exists an integer $a>1$ such that:

(3.1) $a^{N-1}\equiv 1\pmod{N}.$

Then $N$ is prime.

###### Proof.

Since $2K<p$ then $j=1$ will satisfy the condition $p^{j}\geq
p^{2-j}\geq(N-1)^{1/3}$, and since $2K<\log_{a}(2Kp^{2}+1)$ then we will have
$a^{2Kp^{n-j-1}}=a^{2Kp^{2-1-1}}=a^{2K}\equiv L\neq 1\pmod{N}$, also we have
$a^{N-1}\equiv 1\pmod{N}$. Thus all the conditions of Theorem 3.1 are
satisfied and $N$ is prime. ∎

Note that $a=2$ is the best choice in 3.2, since then we will have many
numbers $K$ satisfying the condition $2K<\log_{a}(2Kp^{2}+1)$. The same can be
said as for 2.4 and 3.5.

Now, we will proceed to the second test which requires the following lemma.

######  Lemma 3.3.

If $X>11+4\sqrt{7}$, $a\leq\frac{X}{2}$, $a\neq 1$, $a\neq\sqrt{X+1}$ and
$\frac{(X-a)(a)+1}{X}=M\in\mathbb{Z}$. then $M>\sqrt{X}$.

###### Proof.

If $\frac{(X-a)(a)+1}{X}\in\mathbb{Z}$ then we should have
$\frac{-a^{2}+1}{X}\in\mathbb{Z}$ as well, therefore we could have:

  * (i)

$\frac{-a^{2}+1}{X}=0\Longrightarrow a=1\Longrightarrow M=1$

  * (ii)

$\frac{-a^{2}+1}{X}=-1\Longrightarrow a=\sqrt{X+1}\Longrightarrow
M=\sqrt{X+1}-1$

  * (iii)

$\frac{-a^{2}+1}{X}\leq-2\Longrightarrow\frac{a^{2}-1}{X}\geq 2$.

In (iii) we have $\frac{a^{2}-1}{X}\geq 2$, but since $X>11+4\sqrt{7}$ then we
must have $a>\sqrt{X}+2$, but for which $X>11+4\sqrt{7}$ and $\frac{X}{2}\geq
a>\sqrt{X}+2$ we will have $\frac{(X-a)(a)+1}{X}>\sqrt{X}$ which proves our
result. ∎

######  Theorem 3.4.

Let $N=Kp^{n}+1\neq p^{3(n-j)}+1$, $N\neq(\sqrt{p^{n-j}+1}-1)p^{3(n-j)}+1$ and
$N\neq H^{3}p^{3(n-j)}+1$, where $p$ is prime, $p^{j}\geq p^{2(n-j)}$,
$p^{n-j}\geq(N-1)^{2/7}$ and $p^{n-j}>11+4\sqrt{7}$. If there exists an
integer $a>1$ such that:

  1. (i)

$a^{Kp^{n-j-1}}\equiv L\neq 1\pmod{N}$, and

  2. (ii)

$L^{p^{j+1}}\equiv 1\pmod{N}$.

Then $N$ is prime.

###### Proof.

As in 3.1, we can deduce that there must exist a prime divisor $q$ of $N$ such
that $p^{n-j}\mid q-1$, therefore, we have two cases:

  1. (i)

$q=N$ and $N$ is prime, or

  2. (ii)

$q<N$ and $N$ is composite.

Now, we will try to exclude (ii) by contradiction. In(ii) we have $q\mid N$,
hence, $\frac{N}{q}=M\in\mathbb{Z}$, but since $q=ap^{n-j}+1$ then we must
have $M=bp^{n-j}+1$ as well, now, if we multiply $q$ by $M$ we will get
$N=Kp^{n}+1=abp^{2(n-j)}+(a+b)p^{n-j}+1\Longrightarrow Kp^{j}=abp^{n-j}+a+b$.
Therefore we must have $p^{n-j}\mid a+b$ and
$p^{n-j}\mid(ab+\frac{a+b}{p^{n-j}})$ which leads us to the following two
cases (let $p^{n-j}=X$ for abbreviation):

  1. (1)

$a,b<X\Longrightarrow a+b=X\Longrightarrow
X\mid(ab+1)\Longrightarrow\frac{ab+1}{X}=\frac{N-1}{X^{3}}$

  2. (2)

$a=AX+c$, $b=X-c\Longrightarrow X\mid(AX^{2}-AcX+cX-c^{2}+A+1)\Longrightarrow
AX-Ac+c+\frac{A+1-c^{2}}{X}=\frac{N-1}{X^{3}}$

In (1) we have $\frac{ab+1}{X}\in\mathbb{Z}$, and by assuming that $a\leq
b\Longrightarrow\frac{(X-a)(a)+1}{X}\in\mathbb{Z}$, therefore and as in 3.3
(since $X>11+4\sqrt{7}$) we can deduce that
$a=1\Longrightarrow\frac{N-1}{X^{3}}=1$ or
$a=\sqrt{X+1}\Longrightarrow\frac{N-1}{X^{3}}=\sqrt{X+1}-1$ or
$a>\sqrt{X}+2\Longrightarrow\frac{(X-a)(a)+1}{X}=\frac{N-1}{X^{3}}>\sqrt{X}$
and the latter is impossible since $X\geq(N-1)^{2/7}$, hence, we have excluded
all the cases of (1) (since we assumed that $N\neq p^{3(n-j)}+1$ and
$N\neq(\sqrt{p^{n-j}+1}-1)p^{3(n-j)}+1$).
In (2) we must have $\frac{A+1-c^{2}}{X}\in\mathbb{Z}$ which leads us to the
following three cases

  1. ($\alpha$)

$A+1-c^{2}\geq X$

which is impossible since $A<\sqrt{X}$.

  1. ($\beta$)

$A+1-c^{2}=0\Longrightarrow c=\sqrt{A+1}$

In this case we will have
$ab=(AX+\sqrt{A+1})(X-\sqrt{A+1})=AX^{2}-A\sqrt{A+1}X+\sqrt{A+1}X-(A+1)=AX(X-\sqrt{A+1})+\sqrt{A+1}X-(A+1)$
which is clearly larger than $(X^{3/2})$, since $\sqrt{A+1}X-(A+1)>0$ and
$AX(X-\sqrt{A+1})>X^{3/2}$ (since $X-\sqrt{A+1}>\sqrt{X}$), hence, we have
excluded this case as well.

  1. ($\gamma$)

$c^{2}-A-1\geq X$

In this case we should have $A+1=c^{2}\pmod{X}=(X-c)^{2}\pmod{X}$, but since
$(X-c)<\sqrt{X}$ then $(X-c)^{2}<X$ and we can deduce that
$A=(X-c)^{2}-1\Longrightarrow\frac{N-1}{X^{3}}=((X-c)^{2}-1)(X-c)+c-\frac{c^{2}-1-((X-c)^{2}-1)}{X}=(X-c)^{3}-X+2c-\frac{c^{2}-1-X^{2}+2cX-c^{2}+1}{X}=(X-c)^{3}$,
and we have excluded that case by the assumption $N\neq H^{3}p^{3(n-j)}+1$.
Hence, we have excluded all the cases of (ii), and then we can deduce that
$q=N$ and $N$ is prime. ∎

As an example to illustrate Theorem 3.4, consider $N=14\cdot
3^{18}+1=5423886847$. Then according to theorems 1.5 and 2.5 we can only have
$j\leq 7$, however taking $1\leq j\leq 7$ and $a=1481700844$ will not satisfy
(i) in neither 1.5 nor 2.5. Also in 3.1 we can only have $j\leq 11$, and
taking $1\leq j\leq 11$ with $a=1481700844$ will not satisfy (i) in 3.1 as
well. But in 3.4 we can have $j\leq 12$, and by taking $j=12$, we obtain
$1481700844^{14\cdot 3^{18-12-1}}\equiv 3256260648\pmod{N}$ and the result
follows from Theorem 3.4. Note that the condition
$N=Kp^{n}+1\neq(\sqrt{p^{n-j}+1}-1)p^{3(n-j)}+1$ in 3.4 is not necessary (it
is not necessary in the following corollary as well), since the only cases we
can have $p^{n-j}+1=\square$ is when $p^{n-j}=2^{3}$ or $3$ whose are smaller
than $11+4\sqrt{7}$.

######  Corollary 3.5.

Let $N=2Kp^{3}+1$, where $p$ is prime, $p>11+4\sqrt{7}$,
$2K<\log_{a}(2Kp^{3}+1)$, $2K<\sqrt{p}$, and $2K$ is not a perfect cube. If
there exists an integer $a>1$ such that:

(3.2) $a^{N-1}\equiv 1\pmod{N}.$

Then $N$ is prime.

###### Proof.

Since $2K<\sqrt{p}$ then taking $j=2$ will satisfy the conditions $p^{j}\geq
p^{2(3-j)}$ and $p^{3-j}\geq(N-1)^{2/7}$, and since $2K<\log_{a}(2Kp^{3}+1)$
then we will have $a^{2Kp^{n-j-1}}=a^{2Kp^{3-2-1}}=a^{2K}\equiv L\neq
1\pmod{N}$, also we have $a^{N-1}\equiv 1\pmod{N}$ and $2K$ is not a perfect
cube. Hence all the conditions of Theorem 3.4 are satisfied and $N$ is prime.
∎

##  4\. A generalization of Sophie Germain and Safe primes

Recall that a prime $p$ is called a _SafePrime_ if $\frac{p-1}{2}$ is also
prime. A prime $q$ is called a _SophieGermainPrime_ if $2q+1$ is prime. Thus
there is a one-to-one correspondence between SafePrimes and
SophieGermainPrimes. SafePrimes are useful cryptographic parameters as they
are resilient against certain attacks, see [1, 6] for more details. In this
section we provide certain classes of primes which generalize those notions.
Our motivation comes from the observation that the key feature of SafePrimes
is that $p-1$ has a “large" prime factor (namely $\frac{p-1}{2}$). We thus
propose the following extension.

######  Definition 4.1.

$a$-SafePrime is a prime number of the form $N=2Kp+1$, where $p$ is prime and
$2K<\log_{a}(2Kp+1)$

Our purpose of this notion is to have the factor $p$ of $N-1$ be “much larger"
than any other factor; in particular it is “much larger" than $\frac{N-1}{p}$.
In the classical definition of SafePrimes we simply have $K=1$ and that
condition is clearly satisfied. Note that the primality of $a$-SafePrimes can
be certified only by proving that $a^{N-1}\equiv 1\pmod{N}$ (according to
2.4).

Now, we will define $a$-SophieGermainPrimes as those primes $p$ for which
$2Kp+1$ is also prime for any $K$ with $2K<\log_{a}(2Kp+1)\Longrightarrow
2K\leq\log_{a}(2Kp)\Longrightarrow\frac{a^{2K}}{2K}\leq p$.

Questions regarding the infinitude and density of $a$-SophieGermainPrimes seem
to be more or less of the same order of difficulty as the corresponding
questions for the classical case, where one mostly relies on conjectures and
heuristics (see [2, 4] for instance), but the distribution of
$a$-SophieGermainPrimes will be addressed in the following section, and we
will give an important conjecture about the distribution of
$2$-SophieGermainPrimes (and $a$-SophieGermainPrimes in general). As for the
density of $a$-SafePrimes we recall Hardy and Littlewood’s Conjecture B [3]
which states that for $K\geq 1$, the number of the prime pairs $(p,p+2K)$,
where $p\leq x$, is asymptotically given by

(4.1) $2C_{K}\frac{x}{\log^{2}(x)}\sim
2C_{K}\int_{2}^{x}\frac{dt}{\log^{2}t},$

where

$\displaystyle{C_{K}=c_{2}\prod_{2<q,q\mid 2K}\frac{q-1}{q-2}}.$

and $c_{2}$ (the so-called _twin prime constant_) is given by

$c_{2}=\prod_{p\geq 3}\frac{p(p-2)}{(p-1)^{2}}\approx 0.66016181.$

The exact same asymptotics can be used to estimate the number of the primes of
the form $2Kp+1$, where $p$ is prime and $p\leq x$. Furthermore (as discussed
in [2] for instance), the following function, which is asymptotic to the right
hand side of (4.1) gives a more accurate estimate, especially for smaller
values of $x$

(4.2) $2C_{K}\int_{2}^{x}\frac{dt}{\log t\log 2Kt}.$

If we assume that the number of primes of the form $2Kp+1$ with $K$ ranging
between $1$ and $n$ and $p\leq x$ is simply asymptotic to the sum of the
individual estimates (as given in (4.1) or (4.2)) then we get the estimates:

(4.3) $2\frac{x}{\log^{2}(x)}(C_{1}+C_{2}+\dots+C_{n})$

For which prime $p$ in the range $\frac{a^{2n}}{2n}\leq
p<\frac{a^{2(n+1)}}{2(n+1)}$ $K$ can take values up to $n$ with the condition
$\frac{a^{2K}}{2K}\leq p$ satisfied. For typographical convenience we set
$f(a,n):=\frac{a^{2n}}{2n}$. Utilizing (4.3) the asymptotic behavior for the
number of $a$-SafePrimes for which $f(a,n)\leq p<f(a,n+1)$ is:

$\alpha_{1}(a,n):=2c_{2}\left(\frac{f(a,{n+1})}{\log^{2}(f(a,n+1))}-\frac{f(a,n)}{\log^{2}(f(a,n))}\right)\sum_{K=1}^{n}\prod_{2<q,q\mid
2K}\frac{q-1}{q-2}.$

Similarly, we can use the integral estimate in (4.2) to obtain the estimate:

$\displaystyle{\alpha_{2}(a,n):=2c_{2}\sum_{K=1}^{n}\prod_{2<q,q\mid
2K}\frac{q-1}{q-2}\int_{f(a,n)}^{f(a,n+1)}\frac{dt}{\log t\log 2Kt}}.$

Table 1 lists samples of the actual count of $2$-SafePrimes for which $p$ is
in the interval $[f(2,n),f(2,n+1)]$ (labeled $2$-SP for brevity), as well as
the corresponding estimates $\alpha_{1}(2,n),\alpha_{2}(2,n)$ for $n$ up to
$14$. We also list the number of $2$-SophieGermainPrimes in those intervals
(labeled $2$-SGP). We can not say that there is a one-to-one correspondence
between $2$-SP and $2$-SGP due to the simple fact that the same $2$-SGP might
give rise to more than one $2$-SP for various values of $K$ (the same can be
said as for correspondence between $a$-SP and $a$-SGP). For example when $n=3$
in the table below, the interval $[f(2,3),f(2,4)]=[2^{6}/6,32]$ contains
exactly $7$ primes; namely $11$ through $31$. Two of them ($19$ and $31$) are
not $2$-SGP, but three others give rise to two $2$-SP each (namely $11$, $13$
and $23$).

$n$ |  $2$-SGP in $[f(2,n),f(2,n+1)]$ |  $2$-SP $2Kp+1$, $p$ in $[f(2,n),f(2,n+1)]$ |  $\alpha_{1}(2,n)$ |  $\alpha_{2}(2,n)$
---|---|---|---|---
1 |  2 |  2 |  $-2.74$ |  1.5
2 |  2 |  2 |  $-0.469$ |  3
3 |  5 |  8 |  4 |  9
4 |  14 |  18 |  14 |  20
5 |  36 |  52 |  44 |  54
6 |  104 |  168 |  148 |  165
7 |  295 |  463 |  450 |  483
8 |  895 |  1414 |  1380 |  1447
9 |  2970 |  4854 |  4724 |  4836
10 |  9496 |  15783 |  15484 |  15672
11 |  30788 |  50832 |  50827 |  51030
12 |  104997 |  177808 |  178920 |  178090
13 |  353357 |  596973 |  602484 |  597141
14 |  1211233 |  2041459 |  2066125 |  2040547
Table 1. Some counts and asymptotics of $2$-SafePrimes

##  5\. The probability that a given prime is $a$-SGP

From the definition of $a$-SGP in the previous section, it is obvious that for
which prime $f(a,n)\leq p\leq f(a,n+1)$, there is a set of numbers of the form
$2Kp+1$, where $1\leq K\leq n$, that if there is at least one prime of them,
then we call $p$ an $a$-SGP, that set of numbers will be denoted by
$S_{p}(n)$, namely, $S_{p}(n)=\\{2p+1,4p+1,6p+1,......,2np+1\\}$.
Now, we can’t declare that the elements of $S_{p}(n)$ are independent, since
if $2p+1$ is divisible by $3$ (for example), then all elements in
$\\{8p+1,14p+1,20p+1....\\}$ will be divisible by $3$ as well, namely, we
can’t declare an element of $S_{p}(n)$ is independent of the remained elements
of $S_{p}(n)$ unless it is not divisible by any prime less than $n$.
Therefore, to get an approximation to the number of independent elements in
$S_{p}(n)$ we should get an approximation to the number of elements in
$S_{p}(n)$ which are coprime to all primes less than $n$ (except for 2, since
$S_{p}(n)$ elements are coprime to 2 already), but since for any $N\in
S_{p}(n)$ the probability that $(N,q)=1$ (such that $3\leq q$, and $q$ is
prime) is $\frac{q-1}{q}$, hence, the probability that $(N,q)=1$ for all
primes $3\leq q\leq n$ is $\displaystyle{\prod_{3\leq q\leq n}\frac{q-1}{q}}$,
therefore, the number of elements in $S_{p}(n)$ which are coprime to all
primes less than $n$ (it is also the number of independent elements in
$S_{p}(n)$) is approximately

$\lvert S_{p}(n)\rvert\prod_{3\leq q\leq n}\frac{q-1}{q}=n\prod_{3\leq q\leq
n}\frac{q-1}{q}\approx n\frac{4e^{-\gamma}}{\log n^{2}}.$

That set of independent numbers will be denoted by $ind$-$S_{p}(n)$. Now
assume that $N=2Kp+1\in ind$-$S_{p}(n)$, then the probability that $N$ is
prime (since $N$ is in the interval $[2f(a,n),2nf(a,n+1)]$ and not divisible
by any prime less than $n$) is about:

$\frac{\int_{2f(a,n)}^{2nf(a,n+1)}\frac{dt}{\log
t}}{(2nf(a,n+1)-2f(a,n))\displaystyle{\prod_{2\leq q\leq n}}\frac{q-1}{q}}$

hence, the probability that $N$ is composite is about:

$1-\frac{\int_{2f(a,n)}^{2nf(a,n+1)}\frac{dt}{\log
t}}{\frac{2e^{-\gamma}}{\log n^{2}}(2nf(a,n+1)-2f(a,n))}$

thus, the probability that all the elements of $ind$-$S_{p}(n)$ are composite
is about:

${\left(1-\frac{\int_{2f(a,n)}^{2nf(a,n+1)}\frac{dt}{\log
t}}{\frac{2e^{-\gamma}}{\log
n^{2}}(2nf(a,n+1)-2f(a,n))}\right)}^{n\frac{4e^{-\gamma}}{\log n^{2}}}$

therefore, the probability that not all the elements of $ind$-$S_{p}(n)$ are
composite is about:

$\alpha_{3}(a,n):=1-{\left(1-\frac{\int_{2f(a,n)}^{2nf(a,n+1)}\frac{dt}{\log
t}}{\frac{2e^{-\gamma}}{\log
n^{2}}(2nf(a,n+1)-2f(a,n))}\right)}^{n\frac{4e^{-\gamma}}{\log n^{2}}}$

consequently, $\alpha_{3}(a,n)$ finds the probability that a given prime
$f(a,n)\leq p\leq f(a,n+1)$ is $a$-SGP. In correspondence to $\alpha_{3}(a,n)$
we obtained a similar estimate by assuming that the elements of $S_{p}(n)$ are
independent (in fact they are not), and surprisingly, this obtained estimate
behaved similarly to $\alpha_{3}(a,n)$:

$\alpha_{4}(a,n):=1-{\left(1-\frac{\int_{2f(a,n)}^{2nf(a,n+1)}\frac{dt}{\log
t}}{\frac{1}{2}(2nf(a,n+1)-2f(a,n))}\right)}^{n}.$

But we should acknowledge that both of $\alpha_{3}(a,n)$ and $\alpha_{4}(a,n)$
failed to expect the number of $a$-SGP for large values of $a$ with small $n$,
which compelled us to obtain the following estimate which depends on Hardy and
Littlewood’s Conjecture:

$\alpha_{5}(a,n):=1-\displaystyle{\prod_{K=1}^{n}}\left(1-\frac{2C_{K}\int_{f(a,n)}^{f(a,n+1)}\frac{dt}{\log
t\log 2Kt}}{\int_{f(a,n)}^{f(a,n+1)}\frac{dt}{\log t}}\right)$

note that if we multiply either $\alpha_{3}(a,n)$, $\alpha_{4}(a,n)$ or
$\alpha_{5}(a,n)$ by the expected number of primes in the interval
$[f(a,n),f(a,n+1)]$ (namely, $\int_{f(a,n)}^{f(a,n+1)}\frac{dt}{\log t}$),
then we will get an approximation to the number of $a$-SGP primes in the
interval $[f(a,n),f(a,n+1)]$.
Table 2 lists some counts and asymptotics of $2$-SGP primes in many different
intervals which shows that the accuracy of either $\alpha_{3}(2,n)$,
$\alpha_{4}(2,n)$ or $\alpha_{5}(2,n)$ is very acceptable.
But we should acknowledge that in some similar problems the data agrees well
with heuristics only when the numbers are small, therefore we tried to test
the accuracy of $\alpha_{3}(2,n)$, $\alpha_{4}(2,n)$ and $\alpha_{5}(2,n)$
when dealing with larger numbers, and surprisingly, their accuracy were very
acceptable even if we are dealing with $100$, $1000$ or $10000$ digits
numbers, check table 3.

n |  $\pi(f(2,n+1))-\pi(f(2,n))$ |  $2$-SGP in the interval $[f(2,n),f(2,n+1)]$ |  $\alpha_{3}(2,n)$ $\int_{f(2,n)}^{f(2,n+1)}\frac{dt}{\log t}$ |  $\alpha_{4}(2,n)$ $\int_{f(2,n)}^{f(2,n+1)}\frac{dt}{\log t}$ |  $\alpha_{5}(2,n)$ $\int_{f(2,n)}^{f(2,n+1)}\frac{dt}{\log t}$
---|---|---|---|---|---
1 |  2 |  2 |  – |  2.2 |  2.5
2 |  2 |  2 |  2.8 |  3 |  2.9
3 |  7 |  5 |  5.9 |  5.9 |  6.3
4 |  15 |  14 |  14.1 |  13.8 |  13.9
5 |  42 |  36 |  37 |  36 |  36
6 |  124 |  104 |  104 |  100 |  102
7 |  372 |  295 |  307 |  296 |  297
8 |  1144 |  895 |  941 |  906 |  892
9 |  3647 |  2970 |  2974 |  2864 |  2874
10 |  11861 |  9496 |  9621 |  9272 |  9284
11 |  39258 |  30788 |  31741 |  30617 |  30197
12 |  132119 |  104997 |  106446 |  102770 |  102823
13 |  450453 |  353357 |  361946 |  349778 |  346471
14 |  1553274 |  1211233 |  1245390 |  1204660 |  1185604
15 |  5411233 |  4304679 |  4329368 |  4191628 |  4233170
16 |  19015798 |  14953724 |  15185847 |  14715731 |  14712906
17 |  67343702 |  52508562 |  53689079 |  52071171 |  51643313
18 |  240139092 |  188600098 |  191152371 |  185542307 |  185655903
19 |  861585192 |  671209186 |  684850472 |  665264341 |  660813007
Table 2. Some counts and asymptotics of $2$-SGP $n$ |  $L$ consecutive primes in $[f(2,n),f(2,n+1)]$ |  $2$-SGP out of $L$ consecutive primes in $[f(2,n),f(2,n+1)]$ | $L\alpha_{3}(2,n)$ | $L\alpha_{4}(2,n)$ | $L\alpha_{5}(2,n)$
---|---|---|---|---|---
$169$ |  $100000$ |  76832 | 76986 | 76460 | 76346
$1666$ |  $10000$ |  7669 | 7646 | 7638 | 7636
$16617$ |  $100$ |  77 | 76 | 76 | 76
Table 3. Some counts and asymptotics of $2$-SGP out of some $100$, $1000$, and
$10000$ digits consecutive primes

The computations in tables 2 and 3 show that the ratio of $2$-SGP to primes
slowly converges to $0.76$, also our computations of $\alpha_{3}(2,n)$,
$\alpha_{4}(2,n)$ and $\alpha_{5}(2,n)$ for some large values of $n$ show that
they converge to $0.76$ which means that the probability that a given prime
$p$ is $2$-SGP converges to $0.76$ as well (as $p\to\infty$). Therefore,
according to our computations of $\alpha_{3}(2,n)$, $\alpha_{4}(2,n)$ and
$\alpha_{5}(2,n)$ and supported by the computations in tables 2 and 3, we
believe that it is reasonable to give the following conjecture

######  Conjecture 5.1.

The probability that a given prime $p$ is $2$-SGP converges to $0.76$ as
$p\to\infty$.

We also computed $\alpha_{3}(a,n)$, $\alpha_{4}(a,n)$ and $\alpha_{5}(a,n)$
for some large values of $n$ with many values of $a$ (other than $a=2$), and
according to these computations and supported by the computations in table 4
we give a general conjecture about the probability that a given prime $p$ is
$a$-SGP

######  Conjecture 5.2.

The probability that a given prime $p$ is $a$-SGP converges to
$\displaystyle{\lim_{n\to\infty}\alpha_{3}(a,n)\approx}$
$\displaystyle{\lim_{n\to\infty}\alpha_{4}(a,n)}\approx\lim_{n\to\infty}\alpha_{5}(a,n)$
as $p\to\infty$.

$(a,n)$ |  $L$ consecutive primes in $[f(a,n),f(a,n+1)]$ |  $a$-SGP out of $L$ consecutive primes in $[f(a,n),f(a,n+1)]$ | $L\alpha_{3}(a,n)$ | $L\alpha_{4}(a,n)$ | $L\alpha_{5}(a,n)$
---|---|---|---|---|---
$(2,169)$ |  $100000$ |  76832 | 76986 | 76460 | 76346
$(3,106)$ |  $100000$ |  59943 | 60230 | 59727 | 59555
$(4,84)$ |  $100000$ |  51493 | 51724 | 51276 | 50998
$(5,72)$ |  $100000$ |  46186 | 46512 | 46104 | 45798
$(6,65)$ |  $100000$ |  42743 | 42932 | 42557 | 42077
$(7,59)$ |  $100000$ |  39449 | 40288 | 39934 | 39239
$(8,56)$ |  $100000$ |  37924 | 38236 | 37906 | 37316
$(9,53)$ |  $100000$ |  36224 | 36585 | 36271 | 35606
$(10,50)$ |  $100000$ |  34702 | 35217 | 34915 | 34332
Table 4. Some counts and asymptotics of $a$-SGP out of $10^{5}$ $100$ digits
consecutive primes

To show the utility of 5.1 we considered the question of which of the Mersenne
primes (i.e. those of the form $2^{p}-1$ where $p$ itself is prime) are also
$2$-SGP. Either $\alpha_{3}(2,n)$, $\alpha_{4}(2,n)$ or $\alpha_{5}(2,n)$ (or
according to 5.1 ) can tell us that there are about 23 $2$-SGP out of the
first 30 Mersenne primes, if that was true then we will be able to generate 23
$2$-SP at least. But, out of the first 30 Mersenne primes there are exactly 21
$2$-SGP, which shows again that the accuracy of either $\alpha_{3}(2,n)$,
$\alpha_{4}(2,n)$ or $\alpha_{5}(2,n)$ (or 5.1) is acceptable (even if we are
dealing with completely random numbers such as Mersenne primes) and we can use
them as a method to search for huge $2$-SP or $a$-SP in general. The following
table lists all of the 21 Mersenne primes which are also $2$-SGP as well as
the corresponding values of $2K$ for each one (for some of them, more than one
value of $K$ works).

$N=2K\times(2^{p}-1)+1$ | digits | $N=2K\times(2^{p}-1)+1$ | digits
---|---|---|---
$2\times(2^{2}-1)+1$ | 1 | $2010\times(2^{4253}-1)+1$ | 1284
$4\times(2^{3}-1)+1$ | 2 | $3708\times(2^{4253}-1)+1$ | 1284
$4\times(2^{7}-1)+1$ | 3 | $1746\times(2^{9689}-1)+1$ | 2920
$12\times(2^{17}-1)+1$ | 7 | $3426\times(2^{9941}-1)+1$ | 2997
$16\times(2^{19}-1)+1$ | 7 | $3696\times(2^{9941}-1)+1$ | 2997
$52\times(2^{61}-1)+1$ | 21 | $6238\times(2^{11213}-1)+1$ | 3380
$66\times(2^{61}-1)+1$ | 21 | $9048\times(2^{11213}-1)+1$ | 3380
$114\times(2^{127}-1)+1$ | 41 | $858\times(2^{21701}-1)+1$ | 6536
$124\times(2^{127}-1)+1$ | 41 | $14712\times(2^{21701}-1)+1$ | 6537
$336\times(2^{521}-1)+1$ | 160 | $4018\times(2^{23209}-1)+1$ | 6991
$154\times(2^{607}-1)+1$ | 185 | $20808\times(2^{23209}-1)+1$ | 6991
$550\times(2^{607}-1)+1$ | 186 | $17262\times(2^{44497}-1)+1$ | 13400
$156\times(2^{2203}-1)+1$ | 666 | $15418\times(2^{86243}-1)+1$ | 25966
$546\times(2^{2203}-1)+1$ | 666 | $42844\times(2^{86243}-1)+1$ | 25967
$1110\times(2^{2203}-1)+1$ | 667 | $58818\times(2^{86243}-1)+1$ | 25967
$1144\times(2^{2203}-1)+1$ | 667 | 6526 $\times(2^{132049}-1)+1$ | 39755
$1086\times(2^{2281}-1)+1$ | 690 | 30690 $\times(2^{132049}-1)+1$ | 39756
$1656\times(2^{2281}-1)+1$ | 690 | 47142 $\times(2^{132049}-1)+1$ | 39756
$1816\times(2^{3217}-1)+1$ | 972 | 110086 $\times(2^{132049}-1)+1$ | 39756
Table 5. Mersenne primes which are also $2$-SophieGermainPrimes, and the
corresponding values of $K$

We also considered the question of what is the probability that the largest
known prime $p=2^{82589933}-1$ is $2$-SGP, that probability can be found by
computing either $\alpha_{3}(2,n)$, $\alpha_{4}(2,n)$ or $\alpha_{5}(2,n)$ for
$n=41294979$, hence, the probability that $p=2^{82589933}-1$ is $2$-SGP is
about
$\alpha_{3}(2,41294979)\approx\alpha_{4}(2,41294979)\approx\alpha_{5}(2,41294979)\approx
0.7637$, which means that the probability to find a prime number of the form
$N=2K(2^{82589933}-1)+1$, where $1\leq K\leq 41294979$ is about $0.7637$ .

## Acknowledgment

I am grateful to my supervisors Prof. Ahmed El-Guindy of Cairo University and
Prof. Hatem M.Bahig of Ain Shams University for all their help and guidance
that they have given me. Also I would like to thank Prof. Wafik Lotfallah of
the American University in Cairo and Ms. Hagar Gamal of Cairo University for
several useful discussions.

## References

  * [1] A.J. Menezes, P.C. van Oorschot, and S.A. Vanstone, _Handbook of Applied Cryptography_ , CRC Press, Boca Raton, FL, 1997.
  * [2] Chris K. Caldwell, _An amazing prime heuristic_ , preprint available online at https://arxiv.org/abs/2103.04483v1 1997\.
  * [3] G.H. Hardy, J.E. Littlewood, _Some problems of ‘partitio numerorum’ III. On the expression of a number as a sum of primes_ , Acta Math. 44 (1923) 1-70.
  * [4] J. Korevaar, _The prime pair conjecture of Hardy and Littlewood_ , Indagationes Mathematicae 23 (2012), 269-299.
  * [5] Josè María Grau, Antonio M. Oller-Marcén, and D. Sadornil, _A primality test for $Kp^{n}+1$ numbers_, Math. Comp. 84 (2015), 505-512.
  * [6] Paulo Ribenboim, _The New Book Of Prime Number Records_ , 3rd ed., Springer, New York, 1996.
  * [7] Pedro Berrizbeitia, T. G. Berry, and Juan Tena-Ayuso, _A generalization of Proth’s theorem_ , Acta Arith. 110 (2003), 107-115.
  * [8] Richard Crandall and Carl Pomerance,_Prime Numbers: A Computational Perspective_ , 2nd ed., Springer, New York, 2005.
  * [9] Theophile Pepin, _Sur la formule $2^{2^{n}}+1$_, C. R. Acad. Sci. Paris, 85:329-331, 1878.