# A 27 INTEGERS 10 ( 2010 ) , 319 - 334 THE DIVISIBILITY OF a n − b n BY POWERS
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چکیده
For given integers a, b and j ≥ 1 we determine the set R 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 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 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 a,b.
منابع مشابه
THE DIVISIBILITY OF a − b BY POWERS OF n
For given integers a, b and j ≥ 1 we determine the set R a,b of integers n for which a − b is divisible by n . 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 a,b is probably always finite; this seems difficult to prove, ...
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