Euler's sum of powers conjecture
In
The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case n = 2: if then 2 ≥ k.
Although the conjecture holds for the case k = 3 (which follows from Fermat's Last Theorem for the third powers), it was disproved for k = 4 and k = 5. It is unknown whether the conjecture fails or holds for any value k ≥ 6.
Background
Euler was aware of the equality 594 + 1584 = 1334 + 1344 involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 33 + 43 + 53 = 63 or the taxicab number 1729.[1][2] The general solution of the equation is
where a and b are any integers.
Counterexamples
Euler's 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.[3] This was published in a paper comprising just two sentences.[3] A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:
In 1988, Noam Elkies published a method to construct an infinite sequence of counterexamples for the k = 4 case.[4] His smallest counterexample was
A particular case of Elkies' solutions can be reduced to the identity[5][6]
In 1988, Roger Frye found the smallest possible counterexample
Generalizations
In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[8] that if
- ,
where ai ≠ bj are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m + n ≥ k. In the special case m = 1, the conjecture states that if
(under the conditions given above) then n ≥ k − 1.
The special case may be described as the problem of giving a partition of a perfect power into few like powers. For k = 4, 5, 7, 8 and n = k or k − 1, there are many known solutions. Some of these are listed below.
See OEIS: A347773 for more data.
k = 3
- 33 + 43 + 53 = 63 (Plato's number 216)
- This is the case a = 1, b = 0 of Srinivasa Ramanujan's formula[9]
- A cube as the sum of three cubes can also be parameterized in one of two ways:[9]
- The number 2 100 0003 can be expressed as the sum of three cubes in nine different ways.[9]
k = 4
k = 5
(Lander & Parkin, 1966);[10][11][12] (Lander, Parkin, Selfridge, smallest, 1967);[8] (Lander, Parkin, Selfridge, second smallest, 1967);[8] (Sastry, 1934, third smallest).[8]
k = 6
As of 2002, there are no solutions for k = 6 whose final term is ≤ 730000.[13]
k = 7
(M. Dodrill, 1999).[14]
k = 8
(S. Chase, 2000).[15]
See also
- Jacobi–Madden equation
- Prouhet–Tarry–Escott problem
- Beal's conjecture
- Pythagorean quadruple
- Generalized taxicab number
- Sums of powers, a list of related conjectures and theorems
References
- ISBN 978-0-88385-558-4.
- ^ Titus, III, Piezas (2005). "Euler's Extended Conjecture".
- ^ .
- ^ MR 0930224.
- ^ "Elkies' a4+b4+c4 = d4".
- ^ Piezas III, Tito (2010). "Sums of Three Fourth Powers (Part 1)". A Collection of Algebraic Identities. Retrieved April 11, 2022.
- S2CID 58501120
- ^ JSTOR 2003249.
- ^ a b c "MathWorld : Diophantine Equation--3rd Powers".
- ^ Burkard Polster (March 24, 2018). "Euler's and Fermat's last theorems, the Simpsons and CDC6600". YouTube (video). Archived from the original on 2021-12-11. Retrieved 2018-03-24.
- ^ "MathWorld: Diophantine Equation--5th Powers".
- ^ "A Table of Fifth Powers equal to Sums of Five Fifth Powers".
- ^ Giovanni Resta and Jean-Charles Meyrignac (2002). The Smallest Solutions to the Diophantine Equation , Mathematics of Computation, v. 72, p. 1054 (See further work section).
- ^ "MathWorld: Diophantine Equation--7th Powers".
- ^ "MathWorld: Diophantine Equation--8th Powers".
External links
- Tito Piezas III, A Collection of Algebraic Identities Archived 2011-10-01 at the Wayback Machine
- Jaroslaw Wroblewski, Equal Sums of Like Powers
- Ed Pegg Jr., Math Games, Power Sums
- James Waldby, A Table of Fifth Powers equal to a Fifth Power (2009)
- R. Gerbicz, J.-C. Meyrignac, U. Beckert, All solutions of the Diophantine equation a6 + b6 = c6 + d6 + e6 + f6 + g6 for a,b,c,d,e,f,g < 250000 found with a distributed Boinc project
- EulerNet: Computing Minimal 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!