

A179387


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


29



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

1,1


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^3y^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)


LINKS

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


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*)


CROSSREFS

Cf. A179107, A179108, A179109, A179387, A179388
Sequence in context: A290888 A318554 A109173 * A295364 A199774 A235267
Adjacent sequences: A179384 A179385 A179386 * A179388 A179389 A179390


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



