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%I #35 Mar 12 2022 22:42:55
%S 5,65,1729,635318657
%N Smallest number that is sum of 2 positive distinct n-th powers in 2 different ways.
%C 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
%C 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
%C a(5) > 10^33. - _Julien Courties_, Nov 02 2020
%D 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.
%H Christian Boyer, <a href="http://www.multimagie.com/English/SquaresOfCubes.htm">Squares of Cubes</a>.
%H R. L. Ekl, <a href="https://doi.org/10.1090/S0025-5718-98-00979-X">New results in equal sums of like powers</a>, Math. Comp. 67 (1998) 1309-1315, Table 9.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation5thPowers.html">Diophantine Equation--5th Powers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation6thPowers.html">Diophantine Equation--6th Powers</a>
%H Tom Womack, <a href="http://tom.womack.net/maths/dissert_abstract.htm">Equal Sums of Like Powers</a> [blocked link].
%e 5 = 1^1 + 4^1 = 2^1 + 3^1;
%e 65 = 1^2 + 8^2 = 4^2 + 7^2;
%e 1729 = 1^3 + 12^3 = 9^3 + 10^3; etc.
%t (* 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 *)
%Y Cf. A016078.
%K nonn,nice,hard,more
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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