Question

Showing that $\sum_{x \in \mathbb{F}_p} \left(\frac{x^2-1}{p}\right) = -1$, where $\left(\frac{x}{p}\right)$ is the Legendre symbol

score 245 · Answer 1 · 2026-04-16 19:11:45

245

Views

Showing that $\sum_{x \in \mathbb{F}_p} \left(\frac{x^2-1}{p}\right) = -1$, where $\left(\frac{x}{p}\right)$ is the Legendre symbol

Published on 16 Apr 2026 - 19:11

score 953 · Answer 2 · 2026-04-16 12:52:07

953

Views

Evaluate the Legendre symbol $(\frac{14}{p})$ for $p > 2$.

Published on 16 Apr 2026 - 12:52

score 234 · Answer 3 · 2026-04-17 07:29:04

234

Views

Let $p,q$ be odd primes such that $p-q=4a.$ Prove that $\Bigg(\dfrac{a}{p}\Bigg)=\Bigg(\dfrac{a}{q}\Bigg).$

Published on 17 Apr 2026 - 7:29

score 159 · Answer 4 · 2026-04-13 03:06:36

159

Views

The Number of involutory matrices over $\mathbb{Z_p} $

Published on 13 Apr 2026 - 3:06

score 743 · Answer 5 · 2026-04-17 12:52:03

743

Views

If $p \equiv 3 \ (\text{mod} \ 4)$ is a prime, show $(\frac{p-1}{2})! \equiv (-1)^{t} \ (\text{mod} \ p)$

Published on 17 Apr 2026 - 12:52

score 668 · Answer 6 · 2026-03-25 07:44:57

668

Views

Sum of Legendre symbol $\left(\frac{n^2-a}{p}\right)$ (More explanation)

Published on 25 Mar 2026 - 7:44

score 621 · Answer 7 · 2026-03-25 09:28:15

621

Views

Find p primes odds for Legendre symbol $(-3|p)=+1$ or $(-3|p)=-1$

Published on 25 Mar 2026 - 9:28

score 837 · Answer 8 · 2026-03-25 07:44:41

837

Views

Proof that $5$ is a quadratic residue $(\mod p)$ with $p$ odd prime iif $p \equiv \pm 1 \mod 10$

Published on 25 Mar 2026 - 7:44

score 149 · Answer 9 · 2026-03-25 07:44:00

149

Views

If $p=a^2+b^2$ prove these consequences about $\big(\!\frac{a}{p}\!\big)$

Published on 25 Mar 2026 - 7:44

score 409 · Answer 10 · 2026-03-25 11:07:00

409

Views

Determine the Legendre symbol of $\left(\frac{14}{p}\right)$

Published on 25 Mar 2026 - 11:07

score 1k · Answer 11 · 2026-03-25 07:47:51

1k

Views

Find all primes $p$ : $x^2 \equiv 13 \pmod p$ has solutions

Published on 25 Mar 2026 - 7:47

score 404 · Answer 12 · 2026-04-28 17:38:22

404

Views

What are the Legendre symbols $\left(\frac{10}{31}\right)$ and $\left(\frac{-15}{43}\right)$?

Published on 28 Apr 2026 - 17:38

score 839 · Answer 13 · 2026-03-25 07:45:02

839

Views

Prove that there are infinitely many prime numbers $p$ such that $\left(\frac{a}{p}\right)=1$ for fixed $a$.

Published on 25 Mar 2026 - 7:45

score 709 · Answer 14 · 2026-03-25 07:42:55

709

Views

A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.

Published on 25 Mar 2026 - 7:42

score 354 · Answer 15 · 2026-03-25 07:45:02

354

Views

Proof that Legendre symbol $\Big(\frac{a}{p}\Big)$ is $a^{\frac{p-1}{2}}$

Published on 25 Mar 2026 - 7:45

List Question

Showing that $\sum_{x \in \mathbb{F}_p} \left(\frac{x^2-1}{p}\right) = -1$, where $\left(\frac{x}{p}\right)$ is the Legendre symbol

Evaluate the Legendre symbol $(\frac{14}{p})$ for $p > 2$.

Let $p,q$ be odd primes such that $p-q=4a.$ Prove that $\Bigg(\dfrac{a}{p}\Bigg)=\Bigg(\dfrac{a}{q}\Bigg).$

The Number of involutory matrices over $\mathbb{Z_p} $

If $p \equiv 3 \ (\text{mod} \ 4)$ is a prime, show $(\frac{p-1}{2})! \equiv (-1)^{t} \ (\text{mod} \ p)$

Sum of Legendre symbol $\left(\frac{n^2-a}{p}\right)$ (More explanation)

Find p primes odds for Legendre symbol $(-3|p)=+1$ or $(-3|p)=-1$

Proof that $5$ is a quadratic residue $(\mod p)$ with $p$ odd prime iif $p \equiv \pm 1 \mod 10$

If $p=a^2+b^2$ prove these consequences about $\big(\!\frac{a}{p}\!\big)$

Determine the Legendre symbol of $\left(\frac{14}{p}\right)$

Find all primes $p$ : $x^2 \equiv 13 \pmod p$ has solutions

What are the Legendre symbols $\left(\frac{10}{31}\right)$ and $\left(\frac{-15}{43}\right)$?

Prove that there are infinitely many prime numbers $p$ such that $\left(\frac{a}{p}\right)=1$ for fixed $a$.

A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.

Proof that Legendre symbol $\Big(\frac{a}{p}\Big)$ is $a^{\frac{p-1}{2}}$

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