|
|
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).
|
|
LINKS
|
|
|
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)
This sequence (m=2, n=7)
|
|
KEYWORD
|
nonn,more
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|