A179388 Values y for records of minima of positive distances d = A179386(n) = A154333(x) = x^3 - y^2.

5, 11, 181, 207, 225, 500, 524, 1586, 13537, 376601, 223063347, 911054064, 16073515093, 22143115844, 29448160810, 1661699554612, 2498973838515, 26588790747913, 27582731314539, 178638660622364

Offset

1,1

Comments

"Records of minima" means values A179386(n)=A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently A181138(y) such that A181138(y') > A181138(y) for all y' > y. See the main entry A179386 for all further considerations. - M. F. Hasler, Sep 30 2013

For d values see A179386, for x values see A179387.

Theorem (Artur Jasinski):

For any positive number x >= A179387(n), the distance between the cube of x and the square of any y (with 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 completely computable there can't exist any such x (or the related y).

Links

Table of n, a(n) for n=1..20.

Formula

A179388(n) = sqrt(A179387(n)^3 - A179386(n)).

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}]; yy (*Artur Jasinski*)

Crossrefs

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

Cf. A181138, A229618, A077116, A106265, A165288.

Sequence in context: A020453 A036932 A162252 * A336623 A229907 A181491

Adjacent sequences: A179385 A179386 A179387 * A179389 A179390 A179391

Keyword

more,nonn,hard

Author

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

Extensions

Edited by M. F. Hasler, Sep 30 2013

Status

approved