List questions from quadratic-residues

Published on 18 Sep 2015 - 0:27

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for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$)

Published on 22 Sep 2015 - 8:25

81

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Distribution of QRs mod p

Published on 25 Mar 2026 - 23:36

88

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For every prime $p$ there exists integer numbers $x,y,z$ such that $(x^2+1)(y^2+1)(z^2+1)\equiv1\pmod p$

Published on 25 Sep 2015 - 3:54

262

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If $3|(a^2 + b^2)$, show that $3|a$ and $3|b$.

Published on 01 Oct 2015 - 18:42

577

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How to construct an isomorphism between two quadratic number fields?

Published on 02 Oct 2015 - 9:31

223

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Proving the existence of consecutive quadratic residues modulo $p>5$ by means of Pell's equation

Published on 03 Oct 2015 - 11:01

42

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Show that if you can find four solutions b of $b^2\equiv a\pmod n$, where $n=p*q$, then you can find $p$, $q$

Published on 04 Oct 2015 - 13:55

27

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Mystical relation of QR's of 7

Published on 07 Oct 2015 - 14:58

1.4k

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let $p$ be prime a prime with $p = 5 \mod 8$. furthermore $r \in (\mathbb{Z}/p\mathbb{Z})^*$ of order 4 and let $a$ be a quadratic residue mod $p$.

Published on 08 Oct 2015 - 16:49

88

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if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$.

Published on 09 Oct 2015 - 9:57

92

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verify if $8x^2 - 2x - 3 \equiv 0 \pmod{75}$ has solutions or not.

Published on 25 Mar 2026 - 19:25

66

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Does Gauss's Lemma hold for odd composite numbers?

Published on 13 Oct 2015 - 16:41

319

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Quadratic extension. Decomposition of primes

Published on 24 Oct 2015 - 10:17

163

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Calculating $\sqrt{-1}$

Published on 26 Mar 2026 - 17:38

List Question

If $p\equiv 3\pmod{4}$ and $p\mid x^2+y^2$, prove $p\mid x,y$.

for every positive integer $k$, there exists an integer $x$ such that $kx^2-1$ is quadratic residue (mod $p$)

Distribution of QRs mod p

For every prime $p$ there exists integer numbers $x,y,z$ such that $(x^2+1)(y^2+1)(z^2+1)\equiv1\pmod p$

If $3|(a^2 + b^2)$, show that $3|a$ and $3|b$.

How to construct an isomorphism between two quadratic number fields?

Proving the existence of consecutive quadratic residues modulo $p>5$ by means of Pell's equation

Show that if you can find four solutions b of $b^2\equiv a\pmod n$, where $n=p*q$, then you can find $p$, $q$

Mystical relation of QR's of 7

let $p$ be prime a prime with $p = 5 \mod 8$. furthermore $r \in (\mathbb{Z}/p\mathbb{Z})^*$ of order 4 and let $a$ be a quadratic residue mod $p$.

if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$.

verify if $8x^2 - 2x - 3 \equiv 0 \pmod{75}$ has solutions or not.

Does Gauss's Lemma hold for odd composite numbers?

Quadratic extension. Decomposition of primes

Calculating $\sqrt{-1}$

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