## 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.

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 language | English |
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Pages (from-to) | 319-334 |

Journal | Integers |

Volume | 10 |

Publication status | Published - 2010 |