A198444 Values x for record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).

1, 2, 5, 23, 27, 73, 96, 104, 396, 404, 432, 686, 723, 735, 1130, 1159, 2019, 2031, 3861, 5310, 18219, 18231, 25592, 25608, 44367, 200141, 213842, 308228, 390615, 390635, 549976, 631544, 1579129, 1657086, 2941211, 2941239, 5523608

Offset

1,2

Comments

Distance d is equal to 0 when x = k^2 and y = k^5.

For d values see A198443.

For y values see A198445.

Conjecture (Artur Jasinski): For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of x such that x <> k^2 and y <> k^5) can't be less than A198443(n).

Links

Table of n, a(n) for n=1..37.

J. Blass, A Note on Diophantine Equation Y^2 + k = X^5, Math. Comp. 1976, Vol. 30, No. 135, pp. 638-640.

A. Bremner, On the Equation Y^2 = X^5 + k, Experimental Mathematics 2008 Vol. 17, No. 3, pp. 371-374.

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; Do[m = Floor[(n^5)^(1/2)] + 1; k = m^2 - n^5; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; vecx

Crossrefs

Cf. A179406, A179407, A179408, A198443, A198445.

Sequence in context: A374402 A070281 A355025 * A019368 A141171 A180535

Adjacent sequences: A198441 A198442 A198443 * A198445 A198446 A198447

Keyword

nonn,hard

Author

Artur Jasinski, Oct 25 2011

Status

approved