Question

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

score 164 · Answer 1 · 2026-03-31 05:56:37

164

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Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

Published on 31 Mar 2026 - 5:56

score 82 · Answer 2 · 2026-04-02 14:06:12

82

Views

Prove that $\gcd(a,b,c) = \gcd(\gcd(a,b),\gcd(a,c))$

Published on 02 Apr 2026 - 14:06

score 79 · Answer 3 · 2024-02-10 09:32:46

79

Views

Prove that $(x^3 - x)(2x^2 + 5x - 3)$ is divisible by $5$ without using induction

Published on 10 Feb 2024 - 9:32

score 69 · Answer 4 · 2024-02-11 19:45:30

69

Views

Some questions about proof that $n \nmid 2^n - 1$ for any natural number $n > 1$

Published on 11 Feb 2024 - 19:45

score 97 · Answer 5 · 2024-02-12 11:56:14

97

Views

If $\vert d\vert\geq 3$, is $-1+\sqrt{d}$ an irreducible element of $\mathbb{Z}[\sqrt{d}]$?

Published on 12 Feb 2024 - 11:56

score 34 · Answer 6 · 2026-03-31 05:53:57

34

Views

solution verification | Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$.

Published on 31 Mar 2026 - 5:53

score 72 · Answer 7 · 2026-04-08 04:38:04

72

Views

Is there a relationship with how many lines are necessary to divide a circle?

Published on 08 Apr 2026 - 4:38

score 132 · Answer 8 · 2024-02-16 12:40:54

132

Views

Find all positive integers satisfying $n \mid 5^n+1 $

Published on 16 Feb 2024 - 12:40

score 130 · Answer 9 · 2024-02-18 00:21:39

130

Views

General method for solving problems like $17 \mid 2x + 3y \Longleftrightarrow 17 \mid 9x + 5y$

Published on 18 Feb 2024 - 0:21

score 302 · Answer 10 · 2024-02-20 15:45:03

302

Views

If $a^n+b^n|(ab)^n+1$ for each $n\in\mathbb{N}$ then $1\in\{a,b\}.$

Published on 20 Feb 2024 - 15:45

score 46 · Answer 11 · 2026-03-26 01:12:36

46

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Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series

Published on 26 Mar 2026 - 1:12

score 420 · Answer 12 · 2024-02-22 23:16:36

420

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Why do so many solutions to $n+1\mid3^n+1$ satisfy $n\equiv27\pmod{72}$?

Published on 22 Feb 2024 - 23:16

score 91 · Answer 13 · 2026-04-06 04:40:39

91

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Show that $a+b+c$ divides at least one of the following numbers: $a^3+b^3-2c^3, b^3+c^3-2a^3, a^3+c^ 3-2b^3$

Published on 06 Apr 2026 - 4:40

score 89 · Answer 14 · 2024-02-24 17:15:29

89

Views

Find $n$ such that $7^k\mid 3^n+5^n-1$

Published on 24 Feb 2024 - 17:15

score 69 · Answer 15 · 2024-02-25 16:18:56

69

Views

Show that for every integer $n$, if $n^3 + n$ is divisible by $3$, then $2n^3 + 1$ is not divisible by $3$

Published on 25 Feb 2024 - 16:18

List Question

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

Prove that $\gcd(a,b,c) = \gcd(\gcd(a,b),\gcd(a,c))$

Prove that $(x^3 - x)(2x^2 + 5x - 3)$ is divisible by $5$ without using induction

Some questions about proof that $n \nmid 2^n - 1$ for any natural number $n > 1$

If $\vert d\vert\geq 3$, is $-1+\sqrt{d}$ an irreducible element of $\mathbb{Z}[\sqrt{d}]$?

solution verification | Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$.

Is there a relationship with how many lines are necessary to divide a circle?

Find all positive integers satisfying $n \mid 5^n+1 $

General method for solving problems like $17 \mid 2x + 3y \Longleftrightarrow 17 \mid 9x + 5y$

If $a^n+b^n|(ab)^n+1$ for each $n\in\mathbb{N}$ then $1\in\{a,b\}.$

Solutions of equations of form $\tau(n+a_i)=\tau(x+a_i)$ where $\tau$ is the number of divisors function and $a_i$ is a diverging series

Why do so many solutions to $n+1\mid3^n+1$ satisfy $n\equiv27\pmod{72}$?

Show that $a+b+c$ divides at least one of the following numbers: $a^3+b^3-2c^3, b^3+c^3-2a^3, a^3+c^ 3-2b^3$

Find $n$ such that $7^k\mid 3^n+5^n-1$

Show that for every integer $n$, if $n^3 + n$ is divisible by $3$, then $2n^3 + 1$ is not divisible by $3$

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