MathLabs

Problem 2

For which pairs of positive integers (a,b)(a,b) is the sequence gcd⁡(an+b,bn+a)\gcd(a^n+b,b^n+a), n=1,2,…n=1,2,\ldots, eventually constant?
Step 1 of 5: The pair (1,1) obviously works
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In plain words

With a=b=1a=b=1, every term of the sequence is trivially the same fixed number, so it is constant from the very first term.

(a,b)=(1,1)  ⟹  gcd⁡(1n+1,1n+1)=2(a,b)=(1,1)\implies\gcd(1^n+1,1^n+1)=2
Detailed analysis

If a=b=1a=b=1, then gcd⁡(an+b,bn+a)=gcd⁡(2,2)=2\gcd(a^n+b,b^n+a)=\gcd(2,2)=2 for every nn, so the sequence is (eventually, indeed always) constant.