Euler (1772ab) conjectured that there are no positive integer solutions to the quartic Diophantine
equation
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 ."
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 ." 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.Referenced
on Wolfram|Alpha
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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