A198443 Conjectured 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).

3, 4, 11, 26, 37, 368, 1828, 2180, 7825, 8177, 8217, 71393, 72481, 75154, 118409, 175485, 203697, 206370, 1049148, 1058224, 1843945, 1846618, 8186369, 8197633, 9600802, 96020524, 169503449, 294638801, 305158594, 305192969, 657099024

Offset: 1

Author

Artur Jasinski, Oct 25 2011

Keywords

• nonn
• hard

Comments

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

Only the values of x < 10^8 have been searched/

For x values see A198444.

For y values see A198445.

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

Crossrefs

Cf. A179406, A179407, A179408, A198445.

Programs

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}]; dd

Formula

a(n) = (A198445(n))^2 - (A198444(n))^5.