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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).
9
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
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
KEYWORD
nonn,uned
AUTHOR
Artur Jasinski, Jul 13 2010
STATUS
approved