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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
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
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
KEYWORD
nonn,more
AUTHOR
T. D. Noe, Feb 22 2010
EXTENSIONS
a(17) from Robert Price, Apr 15 2021
STATUS
approved