Euler's sum of powers conjecture explained

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is itself a th power, then is greater than or equal to :

$a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k$

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case : if

	k
a
	1

	k
a
	2

=b^k,

then .

Although the conjecture holds for the case (which follows from Fermat's Last Theorem for the third powers), it was disproved for and . It is unknown whether the conjecture fails or holds for any value .

Background

Euler was aware of the equality 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 or the taxicab number 1729.^[1] ^[2] The general solution of the equation

	3
x
	4

$\begin x_1 &=\lambda(1-(a-3b)(a^2+3b^2)) \\[2pt] x_2 &=\lambda((a+3b)(a^2+3b^2)-1)\\[2pt] x_3 &=\lambda((a+3b)-(a^2+3b^2)^2)\\[2pt] x_4 &= \lambda((a^2+3b^2)^2-(a-3b))\end$

where, and

{λ}

are any rational numbers.

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 .^[3] This was published in a paper comprising just two sentences. A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: $\begin 144^5 &= 27^5 + 84^5 + 110^5 + 133^5 \\ 14132^5 &= (-220)^5 + 5027^5 + 6237^5 + 14068^5 \\ 85359^5 &= 55^5 + 3183^5 + 28969^5 + 85282^5\end$ (Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

In 1988, Noam Elkies published a method to construct an infinite sequence of counterexamples for the case.^[4] His smallest counterexample was $20615673^4 = 2682440^4 + 15365639^4 + 18796760^4.$

A particular case of Elkies' solutions can be reduced to the identity^[5] ^[6] $(85v^2 + 484v - 313)^4 + (68v^2 - 586v + 10)^4 + (2u)^4 = (357v^2 - 204v + 363)^4,$ where $u^2 = 22030 + 28849v - 56158v^2 + 36941v^3 - 31790v^4.$ This is an elliptic curve with a rational point at . From this initial rational point, one can compute an infinite collection of others. Substituting into the identity and removing common factors gives the numerical example cited above.

In 1988, Roger Frye found the smallest possible counterexample $95800^4 + 217519^4 + 414560^4 = 422481^4$ for 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.

Generalizations

See main article: article and Lander, Parkin, and Selfridge conjecture. In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured^[7] that if

	n
\sum
	i=1

	k
a
	i

	m
\sum
	j=1

	k
b
	j

,where are positive integers for all and, then . In the special case, the conjecture states that if

	n
\sum
	i=1

	k
a
	i

=b^k

(under the conditions given above) then .

The special case may be described as the problem of giving a partition of a perfect power into few like powers. For and or, there are many known solutions. Some of these are listed below.

See for more data.

(Plato's number 216)

This is the case, of Srinivasa Ramanujan's formula^[8] $(3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3$

A cube as the sum of three cubes can also be parameterized in one of two ways:^[8] $\begina^3(a^3+b^3)^3 &= b^3(a^3+b^3)^3+a^3(a^3-2b^3)^3+b^3(2a^3-b^3)^3 \\[6pt]a^3(a^3+2b^3)^3 &= a^3(a^3-b^3)^3+b^3(a^3-b^3)^3+b^3(2a^3+b^3)^3\end$

The number 2 100 000³ can be expressed as the sum of three cubes in nine different ways.^[8]

$\begin 422481^4 &= 95800^4 + 217519^4 + 414560^4 \\[4pt] 353^4 &= 30^4 + 120^4 + 272^4 + 315^4\end$ (R. Frye, 1988);^[4] (R. Norrie, smallest, 1911).^[7]

$\begin 144^5 &= 27^5 + 84^5 + 110^5 + 133^5 \\[2pt] 72^5 &= 19^5 + 43^5 + 46^5 + 47^5 + 67^5 \\[2pt] 94^5 &= 21^5 + 23^5 + 37^5 + 79^5 + 84^5 \\[2pt] 107^5 &= 7^5 + 43^5 + 57^5 + 80^5 + 100^5\end$

(Lander & Parkin, 1966);^[9] ^[10] ^[11] (Lander, Parkin, Selfridge, smallest, 1967);^[7] (Lander, Parkin, Selfridge, second smallest, 1967);^[7] (Sastry, 1934, third smallest).^[7]

As of 2002, there are no solutions for whose final term is ≤ 730000.^[12]

$568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7$

(M. Dodrill, 1999).^[13]

$1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8$

(S. Chase, 2000).^[14]

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 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
- - - Euler's Conjecture at library.thinkquest.org
A simple explanation of Euler's Conjecture at Maths Is Good For You!

Notes and References

Book: Dunham . William . 2007 . The Genius of Euler: Reflections on His Life and Work . The MAA . 978-0-88385-558-4 . 220 .
Web site: 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. . 10.1090/S0002-9904-1966-11654-3 . 72 . 6 . 1079. free .
Elkies . Noam . Noam Elkies . 1988 . On A⁴ + B⁴ + C⁴ = D⁴ . . 10.1090/S0025-5718-1988-0930224-9 . 0930224 . 2008781 . 51 . 184 . 825–835 . free .
Web site: Elkies' a⁴+b⁴+c⁴ = d⁴ .
Book: Piezas III, Tito. 2010. Sums of Three Fourth Powers (Part 1). A Collection of Algebraic Identities. http://sites.google.com/site/tpiezas/014. April 11, 2022.
Lander . L. J. . Parkin . T. R. . Selfridge . J. L. . 1967 . A Survey of Equal Sums of Like Powers . . 10.1090/S0025-5718-1967-0222008-0 . 2003249 . 21 . 99 . 446–459 . free .
Web site: MathWorld : Diophantine Equation--3rd Powers.
Web site: Euler's and Fermat's last theorems, the Simpsons and CDC6600 . https://ghostarchive.org/varchive/youtube/20211211/AO-W5aEJ3Wg. 2021-12-11 . live. Burkard Polster . . March 24, 2018 . Burkard Polster . Mathologer --> . video . 2018-03-24.
Web site: MathWorld: Diophantine Equation--5th Powers.
Web site: 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
a^6+b^6+c^6+d^6+e^6=x^6+y⁶

, Mathematics of Computation, v. 72, p. 1054 (See further work section).
Web site: MathWorld: Diophantine Equation--7th Powers.
Web site: MathWorld: Diophantine Equation--8th Powers.