<s>[INST] Number of Solutions to Diophantine Equation $(a)$ Let $c < 2\pi$ be a positive real number. Show that
there are inﬁnitely many integers $n$ such that the equation
$x^2 + y^2 + z^2 = n$
has at least $c\sqrt n$ integer solutions.
$(b)$ Find a constant $C > 0$ such that there are inﬁnitely many $n$ for
which the equation
$x^5 + y^3 + z^2 = n$
has $\ge Cn^{1/30}$ nonnegative solutions
Note: this is not a homework problem
I am not really sure how to go about this problem. Any hints or a general strategy that can be applied to both problems would be very much appreciated.
 [/INST] HINT:Consider $$F(t):=\sum_{n\le t} \#\{x^a+y^b+z^c=n\} $$ and find its asymptotic formula.
Take your problem (a) as an example.
$$\sum_{n\le t} \#\{x^2+y^2+z^2=n\}=\sum_{x,y,z\in\mathbb{Z}}\mathbb{1}_{\{x^2+y^2+z^2\le t\}}=Area\ of\{x^2+y^2+z^2\le t\}+o(Area\ of\{x^2+y^2+z^2\le t\})=\frac{4}{3}\pi t^\frac{3}{2}+o(t^\frac{3}{2})$$
Assume that $\exists c<2\pi$ with only finite many $n$ have has at least $c\sqrt{n}$ integer solutions. Then $$F(t)\le\sum_{n\le t}c\sqrt{n}+O(1)=\frac{2c}{3}t^\frac{3}{2}+o(t^\frac{3}{2})<\frac{4}{3}\pi t^\frac{3}{2}+o(t^\frac{3}{2})$$
A contradiction.
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