Interesting behaviours of $gcd(a^n+b^n, |a - b|)$

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I was messing around the expression $gcd(a^n+b^n, |a - b|)$, and I found that when you fix $a$ and $b$ and increase $n,$ the expression would always be constant. (Shown below)



For example, Let $a = 4$ and $b = 100$.

When $n = 1,$ $gcd(104, 96) = 4$

but when $n = 2, 3, ldots,$ $gcd(4^n+100^n, 96)$ is always $8$



Another example, let $a = 518$ and $b = 712$.

When $n = 1, 2, 3, 4, 5, ldots$

$gcd(cdots)= 28, 56, 112, 224, 224, ldots$



Can someone help with why this is true? (Or provide any counter example? Can't seem to find any with $a, b < 10^3$)



Thank you.



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number-theory greatest-common-divisor




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locked by quid♦ 2 days ago


This question is locked in view of our policy about contest questions. Questions originating from active contests are locked for the duration of the contest, with answers hidden from view by soft-deletion. Please see the comments below for references to the originating contest.














  • I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
    – lulu
    2 days ago














up vote
4
down vote

favorite
2












I was messing around the expression $gcd(a^n+b^n, |a - b|)$, and I found that when you fix $a$ and $b$ and increase $n,$ the expression would always be constant. (Shown below)



For example, Let $a = 4$ and $b = 100$.

When $n = 1,$ $gcd(104, 96) = 4$

but when $n = 2, 3, ldots,$ $gcd(4^n+100^n, 96)$ is always $8$



Another example, let $a = 518$ and $b = 712$.

When $n = 1, 2, 3, 4, 5, ldots$

$gcd(cdots)= 28, 56, 112, 224, 224, ldots$



Can someone help with why this is true? (Or provide any counter example? Can't seem to find any with $a, b < 10^3$)



Thank you.



Try It Yourself!




number-theory greatest-common-divisor




share|cite













locked by quid♦ 2 days ago


This question is locked in view of our policy about contest questions. Questions originating from active contests are locked for the duration of the contest, with answers hidden from view by soft-deletion. Please see the comments below for references to the originating contest.














  • I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
    – lulu
    2 days ago












up vote
4
down vote

favorite
2









up vote
4
down vote

favorite
2






2





I was messing around the expression $gcd(a^n+b^n, |a - b|)$, and I found that when you fix $a$ and $b$ and increase $n,$ the expression would always be constant. (Shown below)



For example, Let $a = 4$ and $b = 100$.

When $n = 1,$ $gcd(104, 96) = 4$

but when $n = 2, 3, ldots,$ $gcd(4^n+100^n, 96)$ is always $8$



Another example, let $a = 518$ and $b = 712$.

When $n = 1, 2, 3, 4, 5, ldots$

$gcd(cdots)= 28, 56, 112, 224, 224, ldots$



Can someone help with why this is true? (Or provide any counter example? Can't seem to find any with $a, b < 10^3$)



Thank you.



Try It Yourself!




number-theory greatest-common-divisor




share|cite













I was messing around the expression $gcd(a^n+b^n, |a - b|)$, and I found that when you fix $a$ and $b$ and increase $n,$ the expression would always be constant. (Shown below)



For example, Let $a = 4$ and $b = 100$.

When $n = 1,$ $gcd(104, 96) = 4$

but when $n = 2, 3, ldots,$ $gcd(4^n+100^n, 96)$ is always $8$



Another example, let $a = 518$ and $b = 712$.

When $n = 1, 2, 3, 4, 5, ldots$

$gcd(cdots)= 28, 56, 112, 224, 224, ldots$



Can someone help with why this is true? (Or provide any counter example? Can't seem to find any with $a, b < 10^3$)



Thank you.



Try It Yourself!





number-theory greatest-common-divisor





share|cite












share|cite




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locked by quid♦ 2 days ago


This question is locked in view of our policy about contest questions. Questions originating from active contests are locked for the duration of the contest, with answers hidden from view by soft-deletion. Please see the comments below for references to the originating contest.






locked by quid♦ 2 days ago


This question is locked in view of our policy about contest questions. Questions originating from active contests are locked for the duration of the contest, with answers hidden from view by soft-deletion. Please see the comments below for references to the originating contest.













  • I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
    – lulu
    2 days ago
















  • I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
    – lulu
    2 days ago















I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
– lulu
2 days ago




I'm voting to close this question as off-topic because it is taken from a live programming competition, codechef.com/AUG18B/problems/GCDMOD
– lulu
2 days ago















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