Question

$N(s) = 1$ if and only if s is a unit if and only if $s = \pm 1$ or $\pm i$. $s \in \mathbb{Z}[i]$

score 116 · Answer 1 · 2020-01-24 18:34:10

116

Views

$N(s) = 1$ if and only if s is a unit if and only if $s = \pm 1$ or $\pm i$. $s \in \mathbb{Z}[i]$

Published on 24 Jan 2020 - 18:34

score 85 · Answer 2 · 2020-02-08 02:23:15

85

Views

What’s the probability that a random walk on the Gaussian integers reaches a prime before a composite?

Published on 08 Feb 2020 - 2:23

score 95 · Answer 3 · 2020-02-15 01:16:31

95

Views

Can any bijective function over the Gaussian integers be represented as the evaluation of a fixed polynomial with coefficients also in the GIs?

Published on 15 Feb 2020 - 1:16

score 645 · Answer 4 · 2020-02-16 15:02:07

645

Views

Is $f$ irreducible in $\mathbb Z[x], \mathbb Q [x], \mathbb Z[i][x], \mathbb Q [i][x]$

Published on 16 Feb 2020 - 15:02

score 402 · Answer 5 · 2020-02-17 15:58:10

402

Views

if $a+bi$ is prime, $a- bi$ is also prime (Gauss integers) (irreducible)

Published on 17 Feb 2020 - 15:58

score 24 · Answer 6 · 2026-03-25 07:45:55

24

Views

Show that $\sum_{j}\chi_{\pi}^3(j)\zeta^{qj} \equiv \chi_{\pi}(q)g(\overline{\chi_{\pi}}) \pmod q.$

Published on 25 Mar 2026 - 7:45

score 572 · Answer 7 · 2020-03-09 12:46:49

572

Views

Prove that $a^2+b^2=p$ has a unique solution $(a,b)\in\mathbb{Z}_{\ge0}^2$ with $a\le b$, where $p\equiv 1\pmod4$ is prime.

Published on 09 Mar 2020 - 12:46

score 541 · Answer 8 · 2020-03-13 20:27:41

541

Views

Prove using Gaussian primes that there are infinitely many primes numbers in the arithmetic progression 1, 5, 9, 13, 17, 21, ....

Published on 13 Mar 2020 - 20:27

score 126 · Answer 9 · 2020-04-11 00:18:57

126

Views

Does $\zeta(s)=\prod \frac{1}{1-p^{-s}}$ converge for $ \Re(s) >1$ for $p= iq $ (Gaussian prime)?what about $\zeta(2),\zeta(4),\cdots$?

Published on 11 Apr 2020 - 0:18

score 649 · Answer 10 · 2020-04-12 15:07:45

649

Views

the ring $\mathbb{Z}[i]/<2+2i>$

Published on 12 Apr 2020 - 15:07

score 142 · Answer 11 · 2020-04-15 15:36:53

142

Views

Integral solutions to $Nx^3=y^2+1$ by Gaussian integers

Published on 15 Apr 2020 - 15:36

score 205 · Answer 12 · 2020-04-19 11:56:50

205

Views

Find all the possible $x,y \in \mathbb{Z}$ s.t. $1125 = x^2 + y^2$

Published on 19 Apr 2020 - 11:56

score 65 · Answer 13 · 2020-04-19 19:39:04

65

Views

What are the zero divisors of the ring $\mathbb Z_n[i]$, where $n$ has only one prime factor which is congruent to 3 (mod 4)?

Published on 19 Apr 2020 - 19:39

score 169 · Answer 14 · 2020-05-06 12:36:14

169

Views

For $p$ odd, show that $p$ is prime in $\mathbb{Z}[i] \iff x^2+1$ does not have roots in $\mathbb{Z}/p\mathbb{Z}$

Published on 06 May 2020 - 12:36

score 100 · Answer 15 · 2020-05-08 13:47:45

100

Views

For $p$ prime in $\mathbb{Z}[i]$, show $p \equiv 3 \mod 4$ using ring theory.

Published on 08 May 2020 - 13:47

List Question

$N(s) = 1$ if and only if s is a unit if and only if $s = \pm 1$ or $\pm i$. $s \in \mathbb{Z}[i]$

What’s the probability that a random walk on the Gaussian integers reaches a prime before a composite?

Can any bijective function over the Gaussian integers be represented as the evaluation of a fixed polynomial with coefficients also in the GIs?

Is $f$ irreducible in $\mathbb Z[x], \mathbb Q [x], \mathbb Z[i][x], \mathbb Q [i][x]$

if $a+bi$ is prime, $a- bi$ is also prime (Gauss integers) (irreducible)

Show that $\sum_{j}\chi_{\pi}^3(j)\zeta^{qj} \equiv \chi_{\pi}(q)g(\overline{\chi_{\pi}}) \pmod q.$

Prove that $a^2+b^2=p$ has a unique solution $(a,b)\in\mathbb{Z}_{\ge0}^2$ with $a\le b$, where $p\equiv 1\pmod4$ is prime.

Prove using Gaussian primes that there are infinitely many primes numbers in the arithmetic progression 1, 5, 9, 13, 17, 21, ....

Does $\zeta(s)=\prod \frac{1}{1-p^{-s}}$ converge for $ \Re(s) >1$ for $p= iq $ (Gaussian prime)?what about $\zeta(2),\zeta(4),\cdots$?

the ring $\mathbb{Z}[i]/<2+2i>$

Integral solutions to $Nx^3=y^2+1$ by Gaussian integers

Find all the possible $x,y \in \mathbb{Z}$ s.t. $1125 = x^2 + y^2$

What are the zero divisors of the ring $\mathbb Z_n[i]$, where $n$ has only one prime factor which is congruent to 3 (mod 4)?

For $p$ odd, show that $p$ is prime in $\mathbb{Z}[i] \iff x^2+1$ does not have roots in $\mathbb{Z}/p\mathbb{Z}$

For $p$ prime in $\mathbb{Z}[i]$, show $p \equiv 3 \mod 4$ using ring theory.

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