Question

Prove that $\displaystyle \prod_{1\leq a<b\leq \frac{p-1}{2}}\,\left(a^2+b^2\right)\equiv \pm1\pmod{p}$ for a prime $p\equiv 3\pmod{4}$.

score 148 · Answer 1 · 2026-04-14 19:36:25

Prove that $\displaystyle \prod_{1\leq a<b\leq \frac{p-1}{2}}\,\left(a^2+b^2\right)\equiv \pm1\pmod{p}$ for a prime $p\equiv 3\pmod{4}$.

score 252 · Answer 2 · 2026-04-15 04:36:50

252

Views

Prove that $5^{2^n} - 1$ is divisible by $2^{n+1}$ for all $n ≥ 1$

Published on 15 Apr 2026 - 4:36

score 197 · Answer 3 · 2026-04-16 11:06:53

197

Views

Prove that there are no non-zero integers $m$ and $n$ such that $m^2 = 180 n^4$

Published on 16 Apr 2026 - 11:06

score 141 · Answer 4 · 2026-04-17 11:11:50

141

Views

Solving a congruence of the form $a^x = b \pmod m$ without indices or primitive roots

Published on 17 Apr 2026 - 11:11

score 151 · Answer 5 · 2026-04-17 05:24:59

151

Views

Does the sequence $ (2\mathbb{N}+1 )^2 +4 $ find primes particularly well?

Published on 17 Apr 2026 - 5:24

score 153 · Answer 6 · 2026-04-14 16:22:09

153

Views

$\forall k,n \in\mathbb N:k\in n\iff k\subsetneq n$ [Proof Verification and Suggestion for shorter or simpler proof]

Published on 14 Apr 2026 - 16:22

score 184 · Answer 7 · 2026-04-15 19:37:26

184

Views

On the product $\prod_{1\leq a<b\leq \frac{p-1}{2}}\,\left(a^2+b^2\right)$, where $p$ is prime

Published on 15 Apr 2026 - 19:37

score 932 · Answer 8 · 2026-04-11 14:41:31

932

Views

Remainder on division with $22$

Published on 11 Apr 2026 - 14:41

score 544 · Answer 9 · 2026-04-14 20:21:07

544

Views

Prove that for all integers $a$ and $b$, if $a\mid b$ then $a^2\mid 3b^2$

Published on 14 Apr 2026 - 20:21

score 694 · Answer 10 · 2026-04-14 04:26:04

694

Views

Possible range of of values modulo p

Published on 14 Apr 2026 - 4:26

score 54 · Answer 11 · 2026-04-15 16:40:34

54

Views

Computing Linear Congruence -Uncertainty Of Answer

Published on 15 Apr 2026 - 16:40

score 74 · Answer 12 · 2026-04-09 21:35:18

74

Views

Problem using binominal theorem?

Published on 09 Apr 2026 - 21:35

score 113 · Answer 13 · 2026-04-11 02:44:35

113

Views

$<$ is a trichotomic relation on $\mathbb N$

Published on 11 Apr 2026 - 2:44

score 1.6k · Answer 14 · 2026-04-16 12:46:39

1.6k

Views

How many pairs of integers $(x,y)$ satisfy the equation $x^{2}-y^{2}=66$

Published on 16 Apr 2026 - 12:46

score 1.6k · Answer 15 · 2026-04-14 01:32:39

1.6k

Views

Proving that a product of 2 complete residue systems of modulo p is not complete (modulo p)

Published on 14 Apr 2026 - 1:32

List Question

Prove that $\displaystyle \prod_{1\leq a<b\leq \frac{p-1}{2}}\,\left(a^2+b^2\right)\equiv \pm1\pmod{p}$ for a prime $p\equiv 3\pmod{4}$.

Prove that $5^{2^n} - 1$ is divisible by $2^{n+1}$ for all $n ≥ 1$

Prove that there are no non-zero integers $m$ and $n$ such that $m^2 = 180 n^4$

Solving a congruence of the form $a^x = b \pmod m$ without indices or primitive roots

Does the sequence $ (2\mathbb{N}+1 )^2 +4 $ find primes particularly well?

$\forall k,n \in\mathbb N:k\in n\iff k\subsetneq n$ [Proof Verification and Suggestion for shorter or simpler proof]

On the product $\prod_{1\leq a<b\leq \frac{p-1}{2}}\,\left(a^2+b^2\right)$, where $p$ is prime

Remainder on division with $22$

Prove that for all integers $a$ and $b$, if $a\mid b$ then $a^2\mid 3b^2$

Possible range of of values modulo p

Computing Linear Congruence -Uncertainty Of Answer

Problem using binominal theorem?

$<$ is a trichotomic relation on $\mathbb N$

How many pairs of integers $(x,y)$ satisfy the equation $x^{2}-y^{2}=66$

Proving that a product of 2 complete residue systems of modulo p is not complete (modulo p)

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