Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of non-zero integers is itself a kth power, then n is greater than or equal to k.
In symbols, if where and are non-zero integers, then .
The conjecture represents an attempt to generalize Fermat's last theorem, which could be seen as the special case of n = 2: if , then .
Although the conjecture holds for the case of k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It still remains unknown if the conjecture fails or holds for any value k ≥ 6.
Euler had an equality for four fourth powers , this however is not a counterexample. He also provided a complete solution to the four cubes problem as in Plato's number or the taxicab number 1729. The general solution for:
where and are any integers.
k = 5
The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found a counterexample for k = 5. A total of three counterexamples that are primitive (no common factors) are known:
- 275 + 845 + 1105 + 1335 = 1445, (Lander, Parkin, 1966)
- (-220)5 + 50275 + 62375 + 140685 = 141325, (Scher, Seidl, 1996)
- 555 + 31835 + 289695 + 852825 = 853595, (Jim Frye, 2004)
k = 4
- 26824404 + 153656394 + 187967604 = 206156734.
- (85v2+484v−313)4 + (68v2−586v+10)4 + (2u)4 = (357v2−204v+363)4
- u2 = 22030+28849v−56158v2+36941v3−31790v4.
This is an elliptic curve with a rational point at v1 = −31/467. From this initial rational point, one can compute an infinite collection of others. Substituting v1 into the identity and removing common factors gives the numerical example cited above.
- 958004 + 2175194 + 4145604 = 4224814.
Moreover, this solution is the only one with values of the variables below 1,000,000.
In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if , where ai ≠ bj are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m+n ≥ k. This would imply as a special case that if
(under the conditions given above) then n ≥ k−1.
- Lander, Parkin, and Selfridge conjecture
- Jacobi–Madden equation
- Prouhet–Tarry–Escott problem
- Beal's conjecture
- Pythagorean quadruple
- Sums of powers, a list of related conjectures and theorems
- William Dunham, ed. (2007). The Genius of Euler: Reflections on His Life and Work. The MAA. p. 220. ISBN 978-0-88385-558-4.
- Titus Piezas III (2005). "Euler's Extended Conjecture".
- L. J. Lander, T. R. Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72: 1079. doi:10.1090/S0002-9904-1966-11654-3.
- Noam Elkies (1988). "On A4 + B4 + C4 = D4". Mathematics of Computation 51 (184): 825–835. doi:10.2307/2008781. JSTOR 2008781. MR 0930224.
- "Elkies' a^4+b^4+c^4 = d^4".
- "Sums of Three Fourth Powers".
- L. J. Lander, T. R. Parkin, J. L. Selfridge (1967). "A Survey of Equal Sums of Like Powers". Mathematics of Computation 21 (99): 446–459. doi:10.1090/S0025-5718-1967-0222008-0. JSTOR 2003249.
- Tito Piezas III: A Collection of Algebraic Identities
- EulerNet: Computing Minimal Equal Sums Of Like Powers
- Jaroslaw Wroblewski Equal Sums of Like Powers
- Weisstein, Eric W., "Euler's Sum of Powers Conjecture", MathWorld.
- Weisstein, Eric W., "Euler Quartic Conjecture", MathWorld.
- Weisstein, Eric W., "Diophantine Equation--4th Powers", MathWorld.
- Euler's Conjecture at library.thinkquest.org
- A simple explanation of Euler's Conjecture at Maths Is Good For You!
- Mathematicians Find New Solutions To An Ancient Puzzle
- Ed Pegg Jr. Power Sums, Math Games