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Euler Quartic Conjecture

Euler (1772ab) conjectured that there are no positive integer solutions to the quartic Diophantine equation

A^4=B^4+C^4+D^4.

This conjecture was disproved by Elkies (1988), who found an infinite class of solutions.

See also

Diophantine Equation--4th Powers, Euler's Sum of Powers Conjecture

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References

Berndt, B. C. and Bhargava, S. "Ramanujan--For Lowbrows." Amer. Math. Monthly 100, 644-656, 1993.Clay Mathematics Institute. "Birch and Swinnerton-Dyer Conjecture." http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/.Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, p. 648, 2005.Dutch, S. "Power Page: Euler's Conjecture." http://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#eulercon.Elkies, N. "On A^4+B^4+C^4=D^4." Math. Comput. 51, 825-835, 1988.Euler, L. Commentationes Arithmeticae 1, xxxiii, No. 1, 1772a.Euler, L. Commentationes Arithmeticae 2, lxviii, No. 3, 1772b.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 139-140, 1994.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 201, 1998.Jacobi, L. W. and Madden D. J. "On a^4+b^4+c^4+d^4=(a+b+c+d)^4." Amer. Math. Monthly 115, 220-236, 2008.Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of Equal Sums of Like Powers." Math. Comput. 21, 446-459, 1967.Ward, M. "Euler's Problem on Sums of Three Fourth Powers." Duke Math. J. 15, 827-837, 1948.

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Euler Quartic Conjecture

Cite this as:

Weisstein, Eric W. "Euler Quartic Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerQuarticConjecture.html

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