Let $m$ be an integer and $q$ be an odd prime factor of $m^2 + 1$. Is there an obvious reason that $\left(\frac{2m}{q}\right)$ always equals 1? From some numerics, this seems to b …

Prove that doesn't exist $N\in\mathbb{N}$ with property: for all primes $p>N$ exist $n\in{3, 4,\ldots, N}$ such that $n, n-1, n-2$ are quadratic residues modulo $p$.

Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$. Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the stan …

What is the probability that the sum of squares of n randomly chosen numbers from $Z_p$ is a quadratic residue mod p? That is, let $a_1$,..$a_n$ be chosen at random. Then how ofte …

Let $p \equiv 1 \bmod 4$ be a prime number. Define the polynomial $$ f(x) = \sum_{a=1}^{p-1} \Big(\frac{a}{p}\Big) x^a. $$ Then $f(x) = x(1-x)^2(1+x)g(x)$ for some polynomial $g \ …

Hi, I refer to formula (8) in Chapter 1 of H. Davenport, Multiplicative Number Theory, Third Edition, Springer (2000), which says that for primes $q\equiv 3 \bmod 4$: $$ L\left(\l …

Are there any research/books about quadratic residue in other number system.

If $p$ is a prime of the form $4n+3$, the class number $h$ of $Q[\sqrt{-p}]$ can be expressed using the number $V$ of quadratic residues and $N$ nonresidues in the interval $[1,\fr …

For prime p sufficiently large, there is always an integer q such that q is a residue mod p, but neither q−1 nor q+1 are; the number of such residues scales like p/8 (and sim …

It is well known that the upper bound on the number of quadratic residues mod n is approximately n/2 and it reaches this bound for n prime. Is there any similar lower bound on the …

I have been stuck on this problem for days. Given: n = x^2 - a(y^2) (x, z are Integers). Let p be a prime divisor of n. How do I show that the p|x or that the legendre symbo …

Let $\chi$ be a real nonprincipal Dirichlet's character modulo $m$. In my answer to the question on $L(1,\chi)$, I explain a trick for showing that $L(1,\chi)>0$ on the simplest e …

Let $n > 1$ be a square-free natural number, which is fixed. The assertion to be proved is the following: Let $p$ run through primes. Then, $$\left( \frac{n}{p} \right)$$ is eq …

I want to determine if a value is a quadratic residue mod $2^{32}$. I've developed a very fast pre-screening method based on a Bloom Filter that identifies quadratic residues for …