# Euler's sum of powers conjecture

**Euler's conjecture** is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers *n* and *k* greater than 1, if the sum of *n* *k*th powers of positive integers is itself a *k*th power, then *n* is greater than or equal to *k*:

- a
*k*

1 + a*k*

2 + ... + a*k**n*=*b*⇒^{k}*n*≥*k*

The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case *n* = 2: if a *k*

1 + a *k*

2 = *b ^{k}*, 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 59^{4} + 158^{4} = 133^{4} + 134^{4} 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 3^{3} + 4^{3} + 5^{3} = 6^{3} 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.[4] A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:

- 27
^{5}+ 84^{5}+ 110^{5}+ 133^{5}= 144^{5}(Lander & Parkin, 1966), - (−220)
^{5}+ 5027^{5}+ 6237^{5}+ 14068^{5}= 14132^{5}(Scher & Seidl, 1996), and - 55
^{5}+ 3183^{5}+ 28969^{5}+ 85282^{5}= 85359^{5}(Frye, 2004).

- 27

In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the *k* = 4 case.[5] His smallest counterexample was

- 2682440
^{4}+ 15365639^{4}+ 18796760^{4}= 20615673^{4}.

- 2682440

A particular case of Elkies' solutions can be reduced to the identity[6][7]

- (85
*v*^{2}+ 484*v*− 313)^{4}+ (68*v*^{2}− 586*v*+ 10)^{4}+ (2*u*)^{4}= (357*v*^{2}− 204*v*+ 363)^{4}

- (85

where

*u*^{2}= 22030 + 28849*v*− 56158*v*^{2}+ 36941*v*^{3}− 31790*v*^{4}.

This is an elliptic curve with a rational point at *v*_{1} = −31/467. From this initial rational point, one can compute an infinite collection of others. Substituting *v*_{1} into the identity and removing common factors gives the numerical example cited above.

In 1988, Roger Frye found the smallest possible counterexample

- 95800
^{4}+ 217519^{4}+ 414560^{4}= 422481^{4}

- 95800

for *k* = 4 by a direct computer search using techniques suggested by Elkies. This solution is the only one with values of the variables below 1,000,000.[8]

## Generalizations

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[9] that if

- ,

where *a _{i}* ≠

*b*are positive integers for all 1 ≤

_{j}*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. As of 2002, there are no solutions for whose final term is ≤ 730000.[10]

*k* = 3

*k*= 3

- 3
^{3}+ 4^{3}+ 5^{3}= 6^{3}(Plato's number 216)

- 3

- This is the case
*a*=1,*b*=0 of Srinivasa Ramanujan's formula

- A cube as the sum of three cubes can also be parameterized as

- or as

- The number 2 100 000
^{3}can be expressed as the sum of three cubes in nine different ways.[11]

*k* = 4

*k*= 4

- 95800
^{4}+ 217519^{4}+ 414560^{4}= 422481^{4}(R. Frye, 1988)[5]

- 95800

- 30
^{4}+ 120^{4}+ 272^{4}+ 315^{4}= 353^{4}(R. Norrie, 1911)[9]

- 30

This is the smallest solution to the problem by R. Norrie.

*k* = 5

*k*= 5

- 27
^{5}+ 84^{5}+ 110^{5}+ 133^{5}= 144^{5}(Lander & Parkin, 1966)[12]

- 27

- 19
^{5}+ 43^{5}+ 46^{5}+ 47^{5}+ 67^{5}= 72^{5}(Lander, Parkin, Selfridge, smallest, 1967)[9]

- 19

- 7
^{5}+ 43^{5}+ 57^{5}+ 80^{5}+ 100^{5}= 107^{5}(Sastry, 1934, third smallest)[9]

- 7

*k* = 7

*k*= 7

- 127
^{7}+ 258^{7}+ 266^{7}+ 413^{7}+ 430^{7}+ 439^{7}+ 525^{7}= 568^{7}(M. Dodrill, 1999)

- 127

*k* = 8

*k*= 8

- 90
^{8}+ 223^{8}+ 478^{8}+ 524^{8}+ 748^{8}+ 1088^{8}+ 1190^{8}+ 1324^{8}= 1409^{8}(S. Chase, 2000)

- 90

## 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

- Dunham, William, ed. (2007).
*The Genius of Euler: Reflections on His Life and Work*. The MAA. p. 220. ISBN 978-0-88385-558-4. - Titus, III, Piezas (2005). "Euler's Extended Conjecture".
- Lander, L. J.; Parkin, T. R. (1966). "Counterexample to Euler's conjecture on sums of like powers".
*Bull. Amer. Math. Soc*.**72**(6): 1079. doi:10.1090/S0002-9904-1966-11654-3. - Washietl, Stefan (17 June 2016). "The Shortest Papers Ever Published". Retrieved 14 September 2019.
- Elkies, Noam (1988). "On
*A*^{4}+*B*^{4}+*C*^{4}=*D*^{4}" (PDF).*Mathematics of Computation*.**51**(184): 825–835. doi:10.1090/S0025-5718-1988-0930224-9. JSTOR 2008781. MR 0930224. - "Elkies'
*a*^{4}+*b*^{4}+*c*^{4}=*d*^{4}". - "Sums of Three Fourth Powers".
- Frye, Roger E. (1988), "Finding 95800
^{4}+ 217519^{4}+ 414560^{4}= 422481^{4}on the Connection Machine",*Proceedings of Supercomputing 88, Vol.II: Science and Applications*, pp. 106–116, doi:10.1109/SUPERC.1988.74138 - Lander, L. J.; Parkin, T. R.; Selfridge, J. L. (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. - 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). - Math world : Diophantine Equation--3rd Powers
- Burkard Polster (March 24, 2018). "Euler's and Fermat's last theorems, the Simpsons and CDC6600" (video). Retrieved 2018-03-24.

## External links

- Tito Piezas III, A Collection of Algebraic Identities
- 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
*a*^{6}+*b*^{6}=*c*^{6}+*d*^{6}+*e*^{6}+*f*^{6}+*g*^{6}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!