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%I #22 Mar 14 2015 10:11:10
%S 2,50,87539319
%N Smallest number that is the sum of two positive n-th powers in >= n ways.
%C Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums." Thus no one knows whether a(4) exists, which requires four such equal sums.
%C a(4) > 10^21 (if it exists). There is no number <= 10^21 that is the sum of two positive 4th powers in >= three ways. - _Donovan Johnson_, Jan 07 2014
%D R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.
%F a(n) >= A016078(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3.
%e 2 = 1^1 + 1^1.
%e 50 = 1^2 + 7^2 = 5^2 + 5^2.
%e 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.
%Y Cf. A048610, A011541 for a(2), a(3).
%Y Cf. also A016078, A230477.
%K hard,more,nonn,bref
%O 1,1
%A _Jonathan Sondow_, Oct 23 2013
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