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A179386
Records of minima of A154333, difference of a cube minus the next smaller square.
31
2, 4, 7, 26, 28, 47, 49, 60, 63, 174, 207, 307, 7670, 15336, 18589, 22189, 37071, 44678, 63604, 64432
OFFSET
1,1
COMMENTS
"Records of minima" here means values A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently, the range of m(x) = min{ A154333(x') ; x' > x }. - M. F. Hasler, Sep 27 2013
For the associated x values see A179387 (and example).
For the associated values y=max{ y | y^2 < x^3 }, see A179388.
From Artur Jasinski, Jul 13 2010: (Start)
Theorem (*Artur Jasinski*)
For any positive number x >= A179387(n) the distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n).
Proof: The number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable, therefore such x can't exist.
(End)
An equivalent theorem is the following (*Artur Jasinski*): For any positive number x >= 1+A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n+1). - Artur Jasinski, Aug 11 2010
Also: The range of b(n) = min { A181138(m) | m>n }. - M. F. Hasler, Sep 26 2013
Indeed, if k=A154333(x) is a member if this sequence A179386, then also k=A181138(y) for the corresponding y, and since there is no larger x' such that x'^3-y'^3 <= k, there cannot be a larger y' such that k=A181138(y') (since this y' would require a corresponding x' > x). Conversely, the same reasoning holds for "records of minima" in A181138. - M. F. Hasler, Sep 26 and Sep 28 2013
LINKS
J. Calvo, J. Herranz, G. Saez, A new algorithm to search for small nonzero |x^3 - y^2| values, Math. Comp. 78 (2009), 2435-2444.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
J. Gebel, A. Petho, G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064
EXAMPLE
For numbers x > 32, A154333(x) > 7.
For numbers x > 35, A154333(x) > 26.
For numbers x > 37, A154333(x) > 28.
For numbers x > 63, A154333(x) > 47.
For numbers x > 65, A154333(x) > 49.
For numbers x > 136, A154333(x) > 60.
For numbers x > 568, A154333(x) > 63.
For numbers x > 5215, A154333(x) > 174.
For numbers x > 367806, A154333(x) > 207.
For numbers x > 939787, A154333(x) > 307.
MATHEMATICA
max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; min = 10^100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10^100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd
CROSSREFS
KEYWORD
more,nonn,hard
AUTHOR
Artur Jasinski, Jul 13 2010, Aug 03 2010
EXTENSIONS
Edited by M. F. Hasler, Sep 27 2013
STATUS
approved