A003824 Numbers that are the sum of two 4th powers in more than one way (primitive solutions).

635318657, 3262811042, 8657437697, 68899596497, 86409838577, 160961094577, 2094447251857, 4231525221377, 26033514998417, 37860330087137, 61206381799697, 76773963505537, 109737827061041, 155974778565937

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The prime divisors of elements of {a(n)} all appear to be in A045390. - David W. Wilson, May 28 2010

Conjecture: a(n) is congruent to 1,2,10 or 17 mod 24. - Mason Korb, Oct 07 2018

Wells selected a(1), with only about 12 other 9-digit numbers, for his Interesting Numbers book. - Peter Munn, May 14 2023

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

References

• L. E. Dickson, History of The Theory of Numbers, Vol. 2 pp. 644-7, Chelsea NY 1923.
• R. K. Guy, Unsolved Problems in Number Theory, D1.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, p. 191.

Links

• D. Wilson, Table of n, a(n) for n = 1..516 [The b-file was computed from Bernstein's list]
• D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4 = a(n)
• D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
• D. J. Bernstein, sortedsums (contains software for computing this and related sequences)
• Leonhard Euler, Resolutio formulae diophanteae ab(maa+nbb)=cd(mcc+ndd) per numeros rationales, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 13 (1802), pp. 45-63. See p. 47.
• John Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
• Carlos Rivera, Puzzle 103. N = a^4+b^4 = c^4+d^4, The Prime Puzzles and Problems Connection.
• E. Rosenstiel et al., 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, Instit. of Mathem. and Its Applic. Bull. Jul 27 (pp. 155-157) 1991.
• Eric Weisstein's World of Mathematics, Diophantine equations, 4th powers

Crossrefs

Cf. A018786.

Extensions

More terms from David W. Wilson, Aug 15 1996