List questions from congruences

Published on 03 May 2015 - 21:36

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$x^2 \equiv a \mod p$, $\ x^2 \equiv b \mod p$, and $x^2 \equiv ab \mod p$, prove either all three are solvable or exactly one

Published on 04 May 2015 - 4:08

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Given $p$ an odd prime, $x^2\equiv a\pmod{p^2}$ and $(a,p)=1$, how could we know that $(x,p)=1$?

Published on 04 May 2015 - 11:54

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Find the set of primes p for which -3 is quadratic residue mod p

Published on 25 Mar 2026 - 22:05

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Solutions of the Congruence

Published on 04 May 2015 - 21:04

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Understanding a proof showing that for any prime $p$, there are integers $x$ and $y$ such that $p|(x^2+y^2+1)$.

Published on 05 May 2015 - 1:39

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Solving quadratic or higher degree congruence with very large modulus.

Published on 05 May 2015 - 23:09

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What is $3^{43} \bmod {33}$?

Published on 05 May 2015 - 23:58

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Check if $n=m^2$ in $\mathbb F_q$

Published on 06 May 2015 - 16:15

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Calculate possible values of $a^4$ mod $120$.

Published on 06 May 2015 - 20:41

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Solve linear congruence: $ax + b = y \; (mod \; m)$

Published on 07 May 2015 - 13:44

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$5^x \equiv 1520 \pmod {9797}$

Published on 07 May 2015 - 18:54

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Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$.

Published on 08 May 2015 - 4:53

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Backwards proof of Fermat's Little Theorem

Published on 08 May 2015 - 9:38

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Find the following integer $ x $, s.t. $x \equiv 7^{57} \pmod {133}$

Published on 09 May 2015 - 21:45

List Question

Number of solutions to congruences

$x^2 \equiv a \mod p$, $\ x^2 \equiv b \mod p$, and $x^2 \equiv ab \mod p$, prove either all three are solvable or exactly one

Given $p$ an odd prime, $x^2\equiv a\pmod{p^2}$ and $(a,p)=1$, how could we know that $(x,p)=1$?

Find the set of primes p for which -3 is quadratic residue mod p

Solutions of the Congruence

Understanding a proof showing that for any prime $p$, there are integers $x$ and $y$ such that $p|(x^2+y^2+1)$.

Solving quadratic or higher degree congruence with very large modulus.

What is $3^{43} \bmod {33}$?

Check if $n=m^2$ in $\mathbb F_q$

Calculate possible values of $a^4$ mod $120$.

Solve linear congruence: $ax + b = y \; (mod \; m)$

$5^x \equiv 1520 \pmod {9797}$

Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$.

Backwards proof of Fermat's Little Theorem

Find the following integer $ x $, s.t. $x \equiv 7^{57} \pmod {133}$

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