OFFSET
1,1
COMMENTS
Randy Ekl discovered that a number that can be written in two ways as a sum of two fifth powers exceeds 4.01*10^30 and one that can be written in two ways as a sum of two sixth powers exceeds 7.25*10^26. - R. J. Mathar, Sep 07 2017
According to the Mathworld links below, a(5) and a(6), if they exist, exceed 1.02*10^26 and 7.25*10^26, respectively. The page at the SquaresOfCubes link below says Stuart Gascoigne did an exhaustive search and found in Sep 2002 that no a(5) solution less than 3.26*10^32 exists. My exhaustive search has determined that any solutions for n > 5, if they exist, must exceed 2^96 (about 7.92*10^28). - Jon E. Schoenfield, Dec 15 2008
a(5) > 10^33. - Julien Courties, Nov 02 2020
REFERENCES
R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.
LINKS
Christian Boyer, Squares of Cubes.
R. L. Ekl, New results in equal sums of like powers, Math. Comp. 67 (1998) 1309-1315, Table 9.
Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers
Eric Weisstein's World of Mathematics, Diophantine Equation--6th Powers
Tom Womack, Equal Sums of Like Powers [blocked link].
EXAMPLE
5 = 1^1 + 4^1 = 2^1 + 3^1;
65 = 1^2 + 8^2 = 4^2 + 7^2;
1729 = 1^3 + 12^3 = 9^3 + 10^3; etc.
MATHEMATICA
(* This naive program is not convenient for n > 3 *) r[n_, k_] := Reduce[0 < x < y && x^n + y^n == k, {x, y}, Integers]; a[n_] := Catch[ For[ k = 1, True, k++, rk = r[n, k]; If[rk =!= False, If[ Head[rk] == Or && Length[rk] == 2, Print["n = ", n, ", k = ", k]; Throw[k]]]]]; Table[a[n], {n, 1, 3}] (* Jean-François Alcover, Jul 30 2013 *)
CROSSREFS
KEYWORD
nonn,nice,hard,more
AUTHOR
STATUS
approved