A179386 Records of minima of A154333, difference of a cube minus the next smaller square.

2, 4, 7, 26, 28, 47, 49, 60, 63, 174, 207, 307, 7670, 15336, 18589, 22189, 37071, 44678, 63604, 64432

Offset: 1

Author

Artur Jasinski, Jul 13 2010, Aug 03 2010

Keywords

• more
• nonn
• hard

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

Examples

			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.
		

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.
• Noam Elkies, Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction arXiv:math/0005139 [math.NT], 2000.
• 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

Crossrefs

Cf. A179107, A179108, A179109, A179387, A179388.

Programs

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

Extensions

Edited by M. F. Hasler, Sep 27 2013