A179406 Record minima of the positive distance d between the fifth power of a positive integer x and the square of an integer y such that d = x^5 - y^2 (x != k^2 and y != k^5).

7, 19, 60, 341, 47776, 70378, 78846, 115775, 220898, 780231, 2242100, 11889984, 26914479, 50406928, 77146256, 80117392, 284679759, 595974650, 2071791247, 7825152599, 67944824923, 742629277177, 1709838230002, 2676465117663

Offset

1,1

Comments

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

For x values see A179407.

For y values see A179408.

Conjecture (from Artur Jasinski): For any positive number x >= A179407(n), the distance d between the fifth power of x and the square of any y (such that x != k^2 and y != k^5) can't be less than A179406(n).

Links

Table of n, a(n) for n=1..24.

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)]; k = n^5 - m^2; 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, 96001}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd

Crossrefs

Cf. A179107, A179108, A179109, A179386, A179387, A179388, A179407, A179408.

Sequence in context: A155335 A155226 A229442 * A165683 A092359 A259797

Adjacent sequences: A179403 A179404 A179405 * A179407 A179408 A179409

Keyword

nonn,uned

Author

Artur Jasinski, Jul 13 2010

Status

approved