What is $\gcd(5^n+7^n, 5^m+7^m)$?

Asked Jan 17 '22 at 20:05

Active Jan 17 '22 at 20:05

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Let $m,n\in \mathbb N$, what is $\gcd(5^n+7^n, 5^m+7^m)$?

Since the case $m=n$ is trivial, we may assume $m>n$, hence by the Euclidean algorithm we get $m=nq+r$. $$d=\gcd(5^n+7^n, 5^m+7^m) =(5^n+7^n, 5^m+7^m-5^{m-n}(5^n+7^n))$$ $$= (5^n+7^n,7^n(7^{m-n}-5^{m-n}))=(5^n+7^n,7^{m-n}-5^{m-n}).$$ Repeating this $q$ times we get $$d= (5^n+7^n,7^{r}-5^{r}) $$ What’s next ?

asked Jan 17 '22 at 20:05

PNT

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Have you looked at any example? Maybe a pattern emerges. – lulu Jan 17 '22 at 20:11
Wait, in the duplicates $m,n$ coprime. What about here. – Mike Jan 17 '22 at 21:50
1

@Mike It gets much more complicated when they are not coprime. EG If $ 3 \mid m, n$, then $5^3 + 7^3 \mid \gcd$. – Calvin Lin Jan 18 '22 at 01:48
@CalvinLin Actually the noncoprime case very easily reduces to the coprime case - see my answer in the linked dupe. – Bill Dubuque Jan 18 '22 at 08:32

What is $\gcd(5^n+7^n, 5^m+7^m)$?

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