Hello I'm the guy who've proven trivially Dirichlet's theorem (I'm a white male genius) and no one has probably heard of me so here we take an attempt to prove Rudin's conjecture, it appears to have some similarities with the methods I used previously, they are basic and should be good enough.



Some literature may provide the correct intuition such as https://arxiv.org/pdf/1301.5122.pdf for a spontaneous example



The conjecture shortly in it's strongest form asserts that [math] N>6, Q(N)=Q(N; 24, 1) [/math] where one defines [math] Q(N) [/math] to be the maximum of the set [math] \{Q(N; q, a):q, a\geq1\} [/math] of amount of perfect squares in an arithmetic progression [math] Q(N; q, a)=q\mathbb{N}_N+a [/math] where it is indexed over the ring of integers modulo [math] N [/math] denoted by [math] \mathbb{N}_N [/math]

>>15379790 →

We start trivially with taking the intersection with the square integers [math]\mathbb{N}^2\cap Q(N; q, a)[/math] for [math] N>26 [/math]

Notice that this intersection is isomorphic to [math]\mathbb{N}_N\cap\frac{1}{q}\big(\mathbb{N}^2-a\big)=\mathbb{N}_N\cap\frac{1}{q}\big((q\mathbb{N}+t_{q, a})^2-a\big)[/math] it is equal in cardinality and where [math]q>t_{q, a}[/math] is such that [math]\frac{1}{q}(t_{q, a}^2-a)\in\mathbb{N}[/math]

This is important worthless talentless tranny it's not your usual mathy little thread that you must put in the other thread



Please go do worthless studies in engineering and stop BOTHERING ME

It’s not like all math has to go into the cesspit of /mg/

We will the notate the set analytically as follows

[eqn]S(q, a)=\frac{1}{q}\big((q\mathbb{N}+t_{q, a})^2-a\big)=\bigcup_{n\in[0, C(q, a)]\cap\mathbb{N}}{qn^2+2nt_{q, a}+\frac{t_{q, a}^2-a}{q}}[/eqn]

Where we find the boundary [math]C(q, a)=\frac{\sqrt{qN+a}-t_{q, a}}{q}[/math] due to the intersection [math]q\mathbb{N}+t_{q, a}\cap\mathbb{N}_{\sqrt{qN+a}}[/math]

DUE TO THE INTERSECTION [math] q\mathbb{N}+t_{q, a}\cap\mathbb{N}_{\sqrt{qN+a}} [/math]

I though about defining differentiation for intervals or in general for [math]\mathbb{N}_x[/math]

Through some routine calculations I have found one interesting equation that might provide the final proof later on

[eqn] (qN+a)(\eta^2_q-a)=a^2\eta_q [/eqn]

Where we take the union of all [math] \eta_q\in\mathbb{N}_q^2\cap q\mathbb{N}_q-a [/math] over that expression above

If any of you care I've gone to sleep, have fun continuing the proof

Nevermind going to kms after I tackle this conjecture so no sleep today!

Meant [math] N>6 [/math] and forget the equation above there is a different inequality to consider

[eqn] |Q|(q; q, a)=|t_{q, a}|=\frac{n_q\sqrt{qN+a}}{qN+2a} [/eqn]

Notice that the amount of perfect squares in [math]Q[/math] is simply [math] |Q|(N; q, a)=C(q, a)+|t_{q, a}|-1 [/math] where we can actually reduce the boundary from >>15382890

to [math] C(q, a)=\frac{\sqrt{qN+a}-t_{q, a}}{q} =\frac{\sqrt{qN+a}}{q}-\{0, 1\}[/math] due to the fact that the floor function for the integers here depends on [math]0&lt;\frac{t_{q, a}}{q}&lt;1[/math]

Arithmetic mistake, it should be [math] |Q|(N; q, a)=C(q, a)|t_{q, a}| [/math] previous derivations are wrong

Except the one with the boundary simplified (verify!)
