The divisibility of a^n-b^n by powers of n

Christopher Smyth, Kalman Gyory

Research output: Contribution to journalArticlepeer-review

Abstract

For given integers a, b and j ! 1 we determine the set R(j) a,b of integers n for which an − bn is divisible by nj . For j = 1, 2, this set is usually infinite; we determine explicitly the exceptional cases for which a, b the set R(j)
a,b (j = 1, 2) is finite. For j = 2, we use Zsigmondy’s Theorem for this. For j ! 3 and gcd(a, b) = 1, R(j) a,b is probably always finite; this seems difficult to prove, however. We also show that determination of the set of integers n for which an + bn is divisible by nj can be reduced to that of R(j) a,b.
Original languageEnglish
Pages (from-to)319-334
JournalIntegers
Volume10
Publication statusPublished - 2010

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