Show that $\gcd(2^m-1, 2^n-1) = 2^ {\gcd(m,n)} -1$

Asked Oct 31 '11 at 09:21

Active Oct 31 '11 at 12:13

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Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$

I'm trying to figure this out:

Show that for all positive integers $m$ and $n$

$\gcd(2^m-1, 2^n-1) = 2^{\gcd(m,n)} -1$

I appreciate your help, Thanks.

Note: $\gcd$ stands for the greatest common divisor.

edited Apr 13 '17 at 12:21

Community

asked Oct 31 '11 at 09:21

Dave

1

Can you at least show that $2^{\gcd(m,n)}-1$ is a common divisor of $2^m-1$ and $2^n-1$? That’s actually fairly easy to see if you write all three numbers in binary. – Brian M. Scott Oct 31 '11 at 09:47
Here is the general case from another question: http://math.stackexchange.com/questions/7473/prove-that-gcdan-1-am-1-a-gcdn-m-1 – Gbean Oct 31 '11 at 09:54
3

Qiaochu Yuan’s solution of the general case here is probably more readable than those at the question that Gbean found. – Brian M. Scott Oct 31 '11 at 10:03
It may be useful to write $(m,n)$ as $am+bn$ where $a,b$ are integers. – Mark Oct 31 '11 at 10:25
http://math.stackexchange.com/questions/225289 – Bart Michels Feb 21 '15 at 16:30

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