Almost fifth powers in arithmetic progression
نویسندگان
چکیده
منابع مشابه
Powers from five terms in arithmetic progression
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 con...
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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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A celebrated theorem of Erdős and Selfridge [14] states that the product of consecutive positive integers is never a perfect power. A more recent and equally appealing result is one of Darmon and Merel [11] who proved an old conjecture of Dénes to the effect that there do not exist three consecutive nth powers in arithmetic progression, provided n 3. One common generalization of these problems ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2011
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2011.04.009