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%I #14 May 07 2021 00:42:45
%S 12,93,239,4896,4904,6546,7806,9104,20542,35962,43783,96569,616400,
%T 635331,842163,7888432,450177181
%N Numbers x such that 0 < |x^7 - y^2| < x^(5/2) for some number y.
%C 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).
%C a(18) > 10^9. - _Robert Price_, Apr 15 2021
%H F. Beukers and C. L. Stewart, <a href="https://doi.org/10.1016/j.jnt.2009.09.006">Neighboring powers</a>, J. Number Theory, 130 (2010), 660-679.
%t 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]<x^pow, AppendTo[t,x]], {x,lim}]; t]; Solutions[7,2,10^7]
%Y Cf. A078933 (m=2, n=3, Hall's conjecture)
%Y Cf. A116884 (m=2, n=5)
%Y This sequence (m=2, n=7)
%Y Cf. A173349 (m=2, n=9)
%Y Cf. A173350 (m=2, n=11)
%Y Cf. A173351 (m=3, n=4)
%Y Cf. A173352 (m=3, n=5)
%Y Cf. A173353 (m=3, n=7)
%Y Cf. A173354 (m=3, n=8)
%Y Cf. A173355 (m=3, n=10)
%Y Cf. A173356 (m=3, n=11)
%Y Cf. A173357 (m=4, n=5)
%Y Cf. A173358 (m=4, n=7)
%Y Cf. A173359 (m=4, n=9)
%Y Cf. A173360 (m=4, n=11)
%Y Cf. A173361 (m=5, n=6)
%Y Cf. A173362 (m=5, n=7)
%Y Cf. A173363 (m=5, n=8)
%Y Cf. A173364 (m=5, n=9)
%Y Cf. A173365 (m=5, n=11)
%Y Cf. A173366 (m=5, n=12)
%Y Cf. A173367 (m=6, n=7)
%Y Cf. A173368 (m=6, n=11)
%Y Cf. A173369 (m=7, n=8)
%Y Cf. A173370 (m=7, n=9)
%Y Cf. A173371 (m=7, n=10)
%Y Cf. A173372 (m=7, n=11)
%Y Cf. A173373 (m=7, n=12)
%Y Cf. A173374 (m=8, n=9)
%Y Cf. A173375 (m=8, n=11)
%Y Cf. A173376 (m=9, n=10)
%Y Cf. A173377 (m=9, n=11)
%Y Cf. A173378 (m=10, n=11)
%Y Cf. A173379 (m=11, n=12)
%K nonn,more
%O 1,1
%A _T. D. Noe_, Feb 22 2010
%E a(17) from _Robert Price_, Apr 15 2021
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