A173348 Numbers x such that 0 < |x^7 - y^2| < x^(5/2) for some number y.

12, 93, 239, 4896, 4904, 6546, 7806, 9104, 20542, 35962, 43783, 96569, 616400, 635331, 842163, 7888432, 450177181

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]<x^pow, AppendTo[t, x]], {x, lim}]; t]; Solutions[7, 2, 10^7]

Crossrefs

Cf. A078933 (m=2, n=3, Hall's conjecture)

Cf. A116884 (m=2, n=5)

This sequence (m=2, n=7)

Cf. A173349 (m=2, n=9)

Cf. A173350 (m=2, n=11)

Cf. A173351 (m=3, n=4)

Cf. A173352 (m=3, n=5)

Cf. A173353 (m=3, n=7)

Cf. A173354 (m=3, n=8)

Cf. A173355 (m=3, n=10)

Cf. A173356 (m=3, n=11)

Cf. A173357 (m=4, n=5)

Cf. A173358 (m=4, n=7)

Cf. A173359 (m=4, n=9)

Cf. A173360 (m=4, n=11)

Cf. A173361 (m=5, n=6)

Cf. A173362 (m=5, n=7)

Cf. A173363 (m=5, n=8)

Cf. A173364 (m=5, n=9)

Cf. A173365 (m=5, n=11)

Cf. A173366 (m=5, n=12)

Cf. A173367 (m=6, n=7)

Cf. A173368 (m=6, n=11)

Cf. A173369 (m=7, n=8)

Cf. A173370 (m=7, n=9)

Cf. A173371 (m=7, n=10)

Cf. A173372 (m=7, n=11)

Cf. A173373 (m=7, n=12)

Cf. A173374 (m=8, n=9)

Cf. A173375 (m=8, n=11)

Cf. A173376 (m=9, n=10)

Cf. A173377 (m=9, n=11)

Cf. A173378 (m=10, n=11)

Cf. A173379 (m=11, n=12)

Sequence in context: A221291 A044263 A044644 * A294449 A073913 A057410

Adjacent sequences: A173345 A173346 A173347 * A173349 A173350 A173351

Keyword

nonn,more

Author

T. D. Noe, Feb 22 2010

Extensions

a(17) from Robert Price, Apr 15 2021

Status

approved