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 117 · Answer 1 · 2026-03-25 12:50:09

117

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 25 Mar 2026 - 12:50

score 86 · Answer 2 · 2026-03-25 14:26:51

86

Views

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

Published on 25 Mar 2026 - 14:26

score 96 · Answer 3 · 2026-03-25 11:14:48

96

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 25 Mar 2026 - 11:14

score 646 · Answer 4 · 2026-03-25 11:15:54

646

Views

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

Published on 25 Mar 2026 - 11:15

score 403 · Answer 5 · 2026-03-25 11:14:54

403

Views

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

Published on 25 Mar 2026 - 11:14

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 573 · Answer 7 · 2026-03-25 11:16:54

573

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 25 Mar 2026 - 11:16

score 542 · Answer 8 · 2026-03-25 11:14:46

542

Views

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

Published on 25 Mar 2026 - 11:14

score 127 · Answer 9 · 2026-03-25 12:50:50

127

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 25 Mar 2026 - 12:50

score 650 · Answer 10 · 2026-03-25 11:09:40

650

Views

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

Published on 25 Mar 2026 - 11:09

score 143 · Answer 11 · 2026-03-25 14:22:40

143

Views

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

Published on 25 Mar 2026 - 14:22

score 206 · Answer 12 · 2026-03-25 11:15:53

206

Views

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

Published on 25 Mar 2026 - 11:15

score 66 · Answer 13 · 2026-03-25 11:16:20

66

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 25 Mar 2026 - 11:16

score 170 · Answer 14 · 2026-03-25 11:16:49

170

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 25 Mar 2026 - 11:16

score 101 · Answer 15 · 2026-03-25 12:54:06

101

Views

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

Published on 25 Mar 2026 - 12:54

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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