All Questions

I am trying to determine whether there are any integers $x,y,z$ such that $$ 1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1) $$ It is clear that $x$ is odd. We can consider this equation as quadratic ...

Since there are only finitely many residues mod $n$, there is some function $H(n) \le n$ such that for all integers $n>1$, $r$, and $e>H(n)$, if $r$ is an $e$-th power mod $n$ then there is some ...

QUESTION. Are my following conjectures true? How to prove them? Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that $$\left(\frac ap\right)=\left(\frac bp\right)=\...

In a recent preprint, I investigated $$S_p(x):=\prod_{k=1}^{(p-1)/2}(x-e^{2\pi ik^2/p}),$$ where $p$ is an odd prime and $x$ is a root of unity. Motivated by Question 337879 and Question 338325, ...

Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...