Question

Is it true that $ \sum_{k=0}^{(2q+1)n+q} \binom{2\left((2q+1)n+q\right)+1}{2k} 3^k $ is divisible by $\sum_{k=0}^{q} \binom{2q+1}{2k} 3^k $

score 112 · Answer 1 · 2026-03-25 07:45:51

112

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Is it true that $ \sum_{k=0}^{(2q+1)n+q} \binom{2\left((2q+1)n+q\right)+1}{2k} 3^k $ is divisible by $\sum_{k=0}^{q} \binom{2q+1}{2k} 3^k $

Published on 25 Mar 2026 - 7:45

score 329 · Answer 2 · 2026-04-01 15:42:29

329

Views

Sum of $1/(1 - x)$ over roots of unity

Published on 01 Apr 2026 - 15:42

score 724 · Answer 3 · 2026-03-25 14:36:49

724

Views

Let $a,b,c$ form a primitive Pythagorean triple, with $a$ odd. Prove that $(c-b)$ and $(c+b)$ share no common prime factors.

Published on 25 Mar 2026 - 14:36

score 441 · Answer 4 · 2018-02-17 12:18:20

Let $A\subset\Bbb N$ such that for all $a\in A,b\in\Bbb N$, there exists some $c$ such that $a+bc$ is a square. What is $A\cap\{1,2,\dots,2018\}$?

score 108 · Answer 5 · 2026-04-09 02:55:13

108

Views

On near-Pythagorean triples $(n^5-2n^3+2n)^2 + (2n^4-2n^2+1)^2 = n^{10} + 1$

Published on 09 Apr 2026 - 2:55

score 229 · Answer 6 · 2026-04-11 18:11:13

229

Views

Help in showing that a function is multiplicative

Published on 11 Apr 2026 - 18:11

score 322 · Answer 7 · 2026-03-25 03:18:13

322

Views

Prove $15\mid 4^{2n}-1$ without induction

Published on 25 Mar 2026 - 3:18

score 478 · Answer 8 · 2018-02-17 19:04:07

How many two digit numbers are there such that when multiplied by 2, 3, 4, 5, 6, 7, 8 or 9 don't change their sum of digits?

score 520 · Answer 9 · 2026-03-25 01:28:06

520

Views

How many solutions does the equation $x^p\equiv x \pmod{p^n}$ have in $\mathbb{Z}_{p^n}$?

Published on 25 Mar 2026 - 1:28

score 868 · Answer 10 · 2026-04-10 23:48:42

868

Views

How to find the last five digits of $3^{1234567}$?

Published on 10 Apr 2026 - 23:48

score 1.3k · Answer 11 · 2026-03-25 15:52:44

1.3k

Views

if $p$ and $q$ are successive odd primes and $p + q = 2r$ then $r$ is composite,

Published on 25 Mar 2026 - 15:52

score 788 · Answer 12 · 2026-03-25 15:52:59

788

Views

Let $p_n$ denote the $n^{th}$ prime then prove that $p_n<2^n$, where $n≥2$

Published on 25 Mar 2026 - 15:52

score 202 · Answer 13 · 2026-04-09 12:03:51

202

Views

Prove that if $n$ is not divisible by 5 or 2,then there exists a number consisting of ones that is divisible by $n$.

Published on 09 Apr 2026 - 12:03

score 305 · Answer 14 · 2018-02-18 10:00:27

305

Views

Find all positive integral soludions of $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$

Published on 18 Feb 2018 - 10:00

score 313 · Answer 15 · 2026-04-11 06:52:26

313

Views

Compositeness tests for specific classes of generalized repunit numbers

Published on 11 Apr 2026 - 6:52

List Question

Is it true that $ \sum_{k=0}^{(2q+1)n+q} \binom{2\left((2q+1)n+q\right)+1}{2k} 3^k $ is divisible by $\sum_{k=0}^{q} \binom{2q+1}{2k} 3^k $

Sum of $1/(1 - x)$ over roots of unity

Let $a,b,c$ form a primitive Pythagorean triple, with $a$ odd. Prove that $(c-b)$ and $(c+b)$ share no common prime factors.

Let $A\subset\Bbb N$ such that for all $a\in A,b\in\Bbb N$, there exists some $c$ such that $a+bc$ is a square. What is $A\cap\{1,2,\dots,2018\}$?

On near-Pythagorean triples $(n^5-2n^3+2n)^2 + (2n^4-2n^2+1)^2 = n^{10} + 1$

Help in showing that a function is multiplicative

Prove $15\mid 4^{2n}-1$ without induction

How many two digit numbers are there such that when multiplied by 2, 3, 4, 5, 6, 7, 8 or 9 don't change their sum of digits?

How many solutions does the equation $x^p\equiv x \pmod{p^n}$ have in $\mathbb{Z}_{p^n}$?

How to find the last five digits of $3^{1234567}$?

if $p$ and $q$ are successive odd primes and $p + q = 2r$ then $r$ is composite,

Let $p_n$ denote the $n^{th}$ prime then prove that $p_n<2^n$, where $n≥2$

Prove that if $n$ is not divisible by 5 or 2,then there exists a number consisting of ones that is divisible by $n$.

Find all positive integral soludions of $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$

Compositeness tests for specific classes of generalized repunit numbers

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