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%I #45 May 14 2023 11:43:50
%S 635318657,3262811042,8657437697,68899596497,86409838577,160961094577,
%T 2094447251857,4231525221377,26033514998417,37860330087137,
%U 61206381799697,76773963505537,109737827061041,155974778565937
%N Numbers that are the sum of two 4th powers in more than one way (primitive solutions).
%C The prime divisors of elements of {a(n)} all appear to be in A045390. - _David W. Wilson_, May 28 2010
%C Conjecture: a(n) is congruent to 1,2,10 or 17 mod 24. - _Mason Korb_, Oct 07 2018
%C Wells selected a(1), with only about 12 other 9-digit numbers, for his Interesting Numbers book. - _Peter Munn_, May 14 2023
%C Dickson (1923) credited Euler with discovering 635318657 as a term, while Leech (1957) proved that it is the least term. - _Amiram Eldar_, May 14 2023
%D L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 644-7, Chelsea NY 1923.
%D R. K. Guy, Unsolved Problems in Number Theory, D1.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, p. 191.
%H D. Wilson, <a href="/A003824/b003824.txt">Table of n, a(n) for n = 1..516</a> [The b-file was computed from Bernstein's list]
%H D. J. Bernstein, <a href="http://cr.yp.to/sortedsums/two4.1000000">List of 516 primitive solutions p^4 + q^4 = r^4 + s^4 = a(n)</a>
%H D. J. Bernstein, <a href="http://pobox.com/~djb/papers/sortedsums.dvi">Enumerating solutions to p(a) + q(b) = r(c) + s(d)</a>
%H D. J. Bernstein, <a href="http://cr.yp.to/sortedsums.html">sortedsums</a> (contains software for computing this and related sequences)
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/716/">Resolutio formulae diophanteae ab(maa+nbb)=cd(mcc+ndd) per numeros rationales</a>, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 13 (1802), pp. 45-63. See p. 47.
%H John Leech, <a href="http://dx.doi.org/10.1017/S0305004100032850">Some solutions of Diophantine equations</a>, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_103.htm">Puzzle 103. N = a^4+b^4 = c^4+d^4</a>, The Prime Puzzles and Problems Connection.
%H E. Rosenstiel et al., <a href="http://www.cix.co.uk/~rosenstiel/cubes/welcome.htm">The Four Least Solutions in Distinct Positive Integers of the Diophantine Equation s = x^3 + y^3 = z^3 + w^3 = u^3 + v^3 = m^3 + n^3</a>, Instit. of Mathem. and Its Applic. Bull. Jul 27 (pp. 155-157) 1991.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DiophantineEquation4thPowers.html">Diophantine equations, 4th powers</a>
%Y Cf. A018786.
%K nonn
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_, Aug 15 1996
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