Powers from five terms in arithmetic progression
نویسنده
چکیده
has only the solution (n, k, b, y, l) = (48, 3, 6, 140, 2) in positive integers n, k, b, y and l, where k, l ≥ 2, P (b) ≤ k and P (y) > k. Here, P (m) denotes the greatest prime factor of the integer m (where, for completeness, we write P (±1) = 1 and P (0) = ∞). Rather surprisingly, no similar conclusion is available for the frequently studied generalization of this equation to products of consecutive terms in arithmetic progression n(n+ d)(n+ 2d) · · · (n+ (k − 1)d) = by, (2)
منابع مشابه
Perfect powers in arithmetic progression 1 PERFECT POWERS IN ARITHMETIC PROGRESSION. A NOTE ON THE INHOMOGENEOUS CASE
We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.
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تاریخ انتشار 2006