A179387 Values x for "records of minima" of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

3, 5, 32, 35, 37, 63, 65, 136, 568, 5215, 367806, 939787, 6369039, 7885438, 9536129, 140292677, 184151166, 890838663, 912903445, 3171881612

Offset: 1

Author

Artur Jasinski, Jul 12 2010, Jul 13 2010, Aug 03 2010

Keywords

• more
• nonn
• hard

Comments

"Records of minima" means values A154333(x) such that A154333(x') > A154333(x) for all x' > x. See the main entry A179386 for all further considerations. - M. F. Hasler, Sep 30 2013

For d values see A179386; For y values see A179388.

Theorem (Artur Jasinski):

For any positive number x >= 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).

Proof: Because number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and complete computable can't existed such x.

From Artur Jasinski, Aug 11 2010: (Start)

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).

(End)

Crossrefs

Cf. A179107, A179108, A179109, A179387, A179388

Programs

Mathematica

max = 1000; vecd = Table[10100, {n, 1, max}]; vecx = Table[10100, {n, 1, max}]; vecy = Table[10100, {n, 1, max}]; len = 1; min = 10100; 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 = 10100; 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}]; xx (*Artur Jasinski*)

Extensions

Edited by M. F. Hasler, Sep 30 2013