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 A173348 Numbers x such that 0 < |x^7 - y^2| < x^(5/2) for some number y. 31
 12, 93, 239, 4896, 4904, 6546, 7806, 9104, 20542, 35962, 43783, 96569, 616400, 635331, 842163, 7888432, 450177181 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Beukers and Stewart conjecture that for coprime integers n and m with n > m >= 2, and for any c > 0, the inequality 0 < |x^n - y^m| < c*X^(1-1/n-1/m) is true for infinitely many positive integers x and y, where X = max(x^n,y^m). They compute such x for 34 pairs (n,m). Given x, it is easy to compute y = round(x^(n/m)). Their tables have been extended to include all terms < 10^7 (or higher to obtain more terms). a(18) > 10^9. - Robert Price, Apr 15 2021 LINKS Table of n, a(n) for n=1..17. F. Beukers and C. L. Stewart, Neighboring powers, J. Number Theory, 130 (2010), 660-679. MATHEMATICA Solutions[n_, m_, lim_] := Module[{x, y, t={}, pow=n*(1-1/m-1/n)}, Do[y=Round[x^(n/m)]; If[0 < Abs[x^n-y^m]

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Last modified October 2 10:29 EDT 2023. Contains 365833 sequences. (Running on oeis4.)