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 non-zero integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ .

In symbols, the conjecture falsely states that if

\sum _{i=1}^{n}a_{i}^{k}=b^{k}

where $n > 1$ and $a 1, a 2, \dots, a n, b$ are non-zero integers, then $n \geq k$ .

The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case $n = 2$ : if $a 1 k + a 2 k = b k$ , then $2 \geq k$ .

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 $k \geq 6$ .

Background

Euler had an equality for four fourth powers 59⁴ + 158⁴ = 133⁴ + 134⁴; 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 for:

x_{1}^{3}+x_{2}^{3}=x_{3}^{3}+x_{4}^{3}

x_{1}=1-(a-3b)(a^{2}+3b^{2}),x_{2}=(a+3b)(a^{2}+3b^{2})-1

x_{3}=(a+3b)-(a^{2}+3b^{2})^{2},x_{4}=(a^{2}+3b^{2})^{2}-(a-3b)

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] A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known:

275 + 84 5 + 110 5 + 133 5 = 144 5

(Lander & Parkin, 1966),

(-220) 5 + 5027 5 + 6237 5 + 7004140680000000000♠ 14068 5 = 7004141320000000000♠ 14132 5

(Scher & Seidl, 1996), and

555 + 3183 5 + 7004289690000000000♠ 28969 5 + 7004852820000000000♠ 85282 5 = 7004853590000000000♠ 85359 5

(Frye, 2004).

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

7006268244000000000♠ 2682440 4 + 7007153656390000000♠ 15365639 4 + 7007187967600000000♠ 18796760 4 = 7007206156730000000♠ 20615673 4

A particular case of Elkies' solutions can be reduced to the identity^[5]^[6]

(85 v 2 + 484 v - 313) 4 + (68 v 2 - 586 v + 10) 4 + (2 u) 4 = (357 v 2 - 204 v + 363) 4

where

u 2 = 7004220300000000000♠ 22030 + 7004288490000000000♠ 28849 v - 7004561580000000000♠ 56158 v 2 + 7004369410000000000♠ 36941 v 3 - 7004317900000000000♠ 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

7004958000000000000♠ 95800 4 + 7005217519000000000♠ 217519 4 + 7005414560000000000♠ 414560 4 = 7005422481000000000♠ 422481 4

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.^[7]

Generalizations

Main article: Lander, Parkin, and Selfridge conjecture

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured^[8] that if $k > 3$ and

\sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}

where $a i \neq b j$ are positive integers for all $1 \leq i \leq n$ and $1 \leq j \leq m$ , then $m + n \geq k$ . In the special case $m = 1$ , the conjecture states that if

\sum _{i=1}^{n}a_{i}^{k}=b^{k}

(under the conditions given above) then $n \geq k - 1$ .

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. There are no solutions for $k = 6$ where $b \leq 7005272580000000000♠ 272580$ .

$k = 4$

7004958000000000000♠ 95800 4 + 7005217519000000000♠ 217519 4 + 7005414560000000000♠ 414560 4 = 7005422481000000000♠ 422481 4

(R. Frye, 1988)^[4]

304 + 120 4 + 272 4 + 315 4 = 353 4

(R. Norrie, 1911)^[8]

$k = 5$

275 + 84 5 + 110 5 + 133 5 = 144 5

(Lander & Parkin, 1966)

195 + 43 5 + 46 5 + 47 5 + 67 5 = 72 5

(Lander, Parkin, Selfridge, smallest, 1967)^[8]

75 + 43 5 + 57 5 + 80 5 + 100 5 = 107 5

(Sastry, 1934, third smallest)^[8]

$k = 7$

1277 + 258 7 + 266 7 + 413 7 + 430 7 + 439 7 + 525 7 = 568 7

(M. Dodrill, 1999)

$k = 8$

908 + 223 8 + 478 8 + 524 8 + 748 8 + 1088 8 + 1190 8 + 1324 8 = 1409 8

(S. Chase, 2000)

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.
1 2 Elkies, Noam (1988). "On A⁴ + B⁴ + C⁴ = D⁴" (PDF). Mathematics of Computation. 51 (184): 825–835. doi:10.1090/S0025-5718-1988-0930224-9. JSTOR 2008781. MR 0930224.
↑ "Elkies' a⁴+b⁴+c⁴ = d⁴".
↑ "Sums of Three Fourth Powers".
↑ Frye, Roger E. (1988), "Finding 95800⁴ + 217519⁴ + 414560⁴ = 422481⁴ on the Connection Machine", Proceedings of Supercomputing 88, Vol.II: Science and Applications, pp. 106–116, doi:10.1109/SUPERC.1988.74138
1 2 3 4 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.

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
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!

Disproved conjectures

Borsuk's Chinese hypothesis Euler's sum of powers Ganea Hauptvermutung Hirsch Mertens Ono's inequality Pólya Ragsdale Schoen–Yau Seifert Tait's Von Neumann Weyl–Berry

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Euler's sum of powers conjecture

Background

Counterexamples

Generalizations

k = 4

k = 5

k = 7

k = 8

See also