Question

How to prove than $a+b+c = 2^n-1$ and $a^2+b^2+c^2 = (4^n-1)/3$ have integer solutions only with Mersenne exponent or exponents of Mersenne exponent?

score 296 · Answer 1 · 2021-05-30 00:57:49

296

Views

How to prove than $a+b+c = 2^n-1$ and $a^2+b^2+c^2 = (4^n-1)/3$ have integer solutions only with Mersenne exponent or exponents of Mersenne exponent?

Published on 30 May 2021 - 0:57

score 538 · Answer 2 · 2021-07-02 00:09:41

538

Views

Why are the Mersenne numbers so factor-poor?

Published on 02 Jul 2021 - 0:09

score 99 · Answer 3 · 2021-09-15 07:30:56

99

Views

Primality criterion for Mersenne numbers involving Euler's totient function

Published on 15 Sep 2021 - 7:30

score 156 · Answer 4 · 2021-09-16 10:58:07

156

Views

Why do Mersenne numbers work?

Published on 16 Sep 2021 - 10:58

score 345 · Answer 5 · 2021-10-06 11:17:53

345

Views

Show that if $n$ is a positive integer greater than $1$, then the Mersenne number $M_n$ cannot be the power of a positive integer.

Published on 06 Oct 2021 - 11:17

score 246 · Answer 6 · 2021-11-01 05:32:12

246

Views

Is $3$ the only prime that is both a Mersenne prime and a Fermat prime?

Published on 01 Nov 2021 - 5:32

score 102 · Answer 7 · 2021-11-11 19:27:35

102

Views

What can be said about the primality of Zsigmondy Numbers?

Published on 11 Nov 2021 - 19:27

score 66 · Answer 8 · 2022-02-09 05:38:46

66

Views

$n$ is prime if $M_n\equiv 1\pmod{n}$

Published on 09 Feb 2022 - 5:38

score 97 · Answer 9 · 2022-04-02 19:31:56

97

Views

Is this a valid proof concerning Mersenne Primes?

Published on 02 Apr 2022 - 19:31

score 96 · Answer 10 · 2022-04-16 21:26:05

96

Views

Assume that there are infinitely many Mersenne primes, prove there are infinitely many positive integers m and n such that $\phi(m)=\sigma(n)$.

Published on 16 Apr 2022 - 21:26

score 79 · Answer 11 · 2022-06-24 13:47:54

79

Views

A diophantine equation inspired in a conjecture due to Gica and Luca

Published on 24 Jun 2022 - 13:47

score 164 · Answer 12 · 2022-07-15 06:45:53

164

Views

A note on conjecture that all the Mersenne numbers are square-free

Published on 15 Jul 2022 - 6:45

score 204 · Answer 13 · 2022-08-07 20:30:21

204

Views

Condition for $8p+1$ divides $2^p-1$?

Published on 07 Aug 2022 - 20:30

score 119 · Answer 14 · 2022-08-08 07:51:00

119

Views

Expected size of least prime factor of $(16^p-1)/15$

Published on 08 Aug 2022 - 7:51

score 239 · Answer 15 · 2022-08-28 12:46:05

239

Views

Remarks on Lucas–Lehmer test

Published on 28 Aug 2022 - 12:46

List Question

How to prove than $a+b+c = 2^n-1$ and $a^2+b^2+c^2 = (4^n-1)/3$ have integer solutions only with Mersenne exponent or exponents of Mersenne exponent?

Why are the Mersenne numbers so factor-poor?

Primality criterion for Mersenne numbers involving Euler's totient function

Why do Mersenne numbers work?

Show that if $n$ is a positive integer greater than $1$, then the Mersenne number $M_n$ cannot be the power of a positive integer.

Is $3$ the only prime that is both a Mersenne prime and a Fermat prime?

What can be said about the primality of Zsigmondy Numbers?

$n$ is prime if $M_n\equiv 1\pmod{n}$

Is this a valid proof concerning Mersenne Primes?

Assume that there are infinitely many Mersenne primes, prove there are infinitely many positive integers m and n such that $\phi(m)=\sigma(n)$.

A diophantine equation inspired in a conjecture due to Gica and Luca

A note on conjecture that all the Mersenne numbers are square-free

Condition for $8p+1$ divides $2^p-1$?

Expected size of least prime factor of $(16^p-1)/15$

Remarks on Lucas–Lehmer test

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