A046881 Smallest number that is sum of 2 positive distinct n-th powers in 2 different ways.

5, 65, 1729, 635318657

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• nice
• hard
• more

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

Examples

			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.
		

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].

Crossrefs

Cf. A016078.

Programs

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 *)