Sequences that commute with gcd

Asked Aug 09 '17 at 18:36

Active Aug 10 '17 at 02:04

Viewed 194 times

Which sequences of integers $P_n$ satisfy $\gcd(P_m,P_n) = P_{\gcd(m,n)}$ ?

This is true for the Fibonacci numbers.

More generally,

Which sequences of integer polynomials satisfy $\gcd(P_m(x),P_n(x)) = P_{\gcd(m,n)}(x)$ ?

(I think this is true for the Fibonacci polynomials.)

edited Aug 10 '17 at 02:04

asked Aug 09 '17 at 18:36

lhf

Motivated by https://math.stackexchange.com/questions/2387994/prove-the-following-algebra-of-polynomials. – lhf Aug 09 '17 at 18:37
$P_n(x)=x^n-1,$ for one example. – dxiv Aug 09 '17 at 18:45
2

They are called strong divisibility sequences. Searching on that will locate much literature. – Bill Dubuque Aug 09 '17 at 18:51
@divx Yes, e.g. see this answer where I give the addition law for $,f_n = (x^n-y^n)/(x-y).,$ I mention some (strong) divisibility sequences in passing in various answers. – Bill Dubuque Aug 09 '17 at 19:02
@BillDubuque, ah, I didn't know the term "strong divisibility sequence". Thanks. – lhf Aug 09 '17 at 19:22
Possible duplicate of https://math.stackexchange.com/questions/2269874/what-is-the-significance-of-a-divisibility-sequence. – lhf Aug 09 '17 at 19:22
1

@lhf But I mentioned it last year in the prior-linked answer whose comments you participated in! $\ \ $ – Bill Dubuque Aug 09 '17 at 20:13
@BillDubuque I go back to old posts of mine, and don't even understand them! – Daniel Donnelly Aug 09 '17 at 22:27
@lhf what is an application of this identity? – Daniel Donnelly Aug 09 '17 at 22:28
@EyesOnBud See e.g. the linked "various answers" above. – Bill Dubuque Aug 10 '17 at 02:35
Also https://math.stackexchange.com/questions/2145404/the-gcd-operator-commutes-with-functions-defined-by-linear-recurrence-relatio. – lhf Aug 11 '17 at 11:47

0 Answers0