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Euler Quartic Conjecture
Euler (1772ab) conjectured that there are nopositive integer solutions to the quarticDiophantine equation
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This conjecture was disproved by Elkies (1988), who found an infinite class of solutions.
See also
Diophantine Equation--4thPowers,Euler's Sum of Powers ConjectureExplore with Wolfram|Alpha
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References
Berndt, B. C. and Bhargava, S. "Ramanujan--For Lowbrows."Amer. Math. Monthly100, 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
."Math. Comput.51, 825-835, 1988.Euler, L.Commentationes Arithmeticae1, xxxiii, No. 1, 1772a.Euler, L.Commentationes Arithmeticae2, 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
."Amer. Math. Monthly115, 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.Referenced on Wolfram|Alpha
Euler Quartic ConjectureCite this as:
Weisstein, Eric W. "Euler Quartic Conjecture."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/EulerQuarticConjecture.html