Question

Find the primes $p$ such that the equation: $x^{2} + 7x + 39 = 0 $ has a solution modulo $ p $.

score 112 · Answer 1 · 2016-08-03 21:50:31

112

Views

Find the primes $p$ such that the equation: $x^{2} + 7x + 39 = 0 $ has a solution modulo $ p $.

Published on 03 Aug 2016 - 21:50

score 568 · Answer 2 · 2016-08-14 14:47:27

568

Views

Prove that $\sum_{X=0}^{p-1} \left(\frac{X^{2}+A}{p}\right)=-1$

Published on 14 Aug 2016 - 14:47

score 151 · Answer 3 · 2026-03-25 12:49:41

151

Views

Legendre Symbol: Prove that if $(a,p)=1$ and $p$ is an odd prime, then $\sum^p_{n=1}\left(\frac{an+b}{p}\right)=0$.

Published on 25 Mar 2026 - 12:49

score 107 · Answer 4 · 2016-09-22 06:57:01

107

Views

Show that if $n\geq 5$ then $1!+2!+3!+\ldots+n!\equiv 3 \pmod{5}$

Published on 22 Sep 2016 - 6:57

score 76 · Answer 5 · 2026-03-25 12:50:48

76

Views

Show that $p$ is a divisor of numbers $m^2 + 1$, $n^2 + 2$, if and only if it is a divisor of some number of the form $k^4 + 1$.

Published on 25 Mar 2026 - 12:50

score 536 · Answer 6 · 2016-10-04 18:54:02

536

Views

Show that the following sum of legendre symbols is -1.

Published on 04 Oct 2016 - 18:54

score 341 · Answer 7 · 2026-03-25 14:26:01

341

Views

Show that $\left(\frac{2}{p}\right) = \large(-1)^{\frac{p^2-1}{8}\large}$

Published on 25 Mar 2026 - 14:26

score 119 · Answer 8 · 2016-10-14 04:54:45

119

Views

Legendre symbol multiplicativity

Published on 14 Oct 2016 - 4:54

score 155 · Answer 9 · 2016-10-23 14:34:20

155

Views

Trying to prove $p \equiv 3 \pmod 4 \iff \sum\limits_{x\in\mathbb{F}_p} \left(\frac{x^3-x}{p} \right)=0$

Published on 23 Oct 2016 - 14:34

score 73 · Answer 10 · 2016-10-31 04:46:28

73

Views

If prime $p$ divides $m^2+1$ and $n^2+2$, then there exists $k$ such that $p$ divides $k^4+1$.

Published on 31 Oct 2016 - 4:46

score 133 · Answer 11 · 2016-11-03 22:06:53

133

Views

Show that x is a quadratic (non-)residue (mod p) if p ≡ ... (mod y)

Published on 03 Nov 2016 - 22:06

score 268 · Answer 12 · 2016-11-06 15:59:46

268

Views

Legendre symbols as multiplicative homomorphisms in number fields

Published on 06 Nov 2016 - 15:59

score 303 · Answer 13 · 2016-11-15 18:13:34

303

Views

How to prove this sum of those legendre symbols is -1?

Published on 15 Nov 2016 - 18:13

score 157 · Answer 14 · 2016-11-23 11:41:45

157

Views

Show that $x^2+x+23\equiv 0\mod 173$ has a solution $\iff \left(\frac{28}{173}\right) = 1$

Published on 23 Nov 2016 - 11:41

score 84 · Answer 15 · 2016-11-28 01:56:17

84

Views

Understanding why $(\frac{a}{p}) \equiv a^{(p-1)/2} \pmod p$ when $p \mid a$

Published on 28 Nov 2016 - 1:56

List Question

Find the primes $p$ such that the equation: $x^{2} + 7x + 39 = 0 $ has a solution modulo $ p $.

Prove that $\sum_{X=0}^{p-1} \left(\frac{X^{2}+A}{p}\right)=-1$

Legendre Symbol: Prove that if $(a,p)=1$ and $p$ is an odd prime, then $\sum^p_{n=1}\left(\frac{an+b}{p}\right)=0$.

Show that if $n\geq 5$ then $1!+2!+3!+\ldots+n!\equiv 3 \pmod{5}$

Show that $p$ is a divisor of numbers $m^2 + 1$, $n^2 + 2$, if and only if it is a divisor of some number of the form $k^4 + 1$.

Show that the following sum of legendre symbols is -1.

Show that $\left(\frac{2}{p}\right) = \large(-1)^{\frac{p^2-1}{8}\large}$

Legendre symbol multiplicativity

Trying to prove $p \equiv 3 \pmod 4 \iff \sum\limits_{x\in\mathbb{F}_p} \left(\frac{x^3-x}{p} \right)=0$

If prime $p$ divides $m^2+1$ and $n^2+2$, then there exists $k$ such that $p$ divides $k^4+1$.

Show that x is a quadratic (non-)residue (mod p) if p ≡ ... (mod y)

Legendre symbols as multiplicative homomorphisms in number fields

How to prove this sum of those legendre symbols is -1?

Show that $x^2+x+23\equiv 0\mod 173$ has a solution $\iff \left(\frac{28}{173}\right) = 1$

Understanding why $(\frac{a}{p}) \equiv a^{(p-1)/2} \pmod p$ when $p \mid a$

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