Euler's sum of powers conjecture

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 n and k greater than 1, if the sum of n many kth powers of positive integers is itself a kth power, then n is greater than or equal to k:

${\displaystyle a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}=b^{k}\implies n\geq k}$

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case n = 2: if ${\displaystyle a_{1}^{k}+a_{2}^{k}=b^{k},}$ 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 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 ${\displaystyle x_{1}^{3}+x_{2}^{3}=x_{3}^{3}+x_{4}^{3}}$ is

{\displaystyle {\begin{aligned}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{aligned}}}

where a, b and ${\displaystyle {\lambda }}$ 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 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: {\displaystyle {\begin{aligned}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{aligned}}} (Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

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

A particular case of Elkies' solutions can be reduced to the identity[5][6] ${\displaystyle (85v^{2}+484v-313)^{4}+(68v^{2}-586v+10)^{4}+(2u)^{4}=(357v^{2}-204v+363)^{4},}$ where ${\displaystyle u^{2}=22030+28849v-56158v^{2}+36941v^{3}-31790v^{4}.}$ 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.

In 1988, Roger Frye found the smallest possible counterexample ${\displaystyle 95800^{4}+217519^{4}+414560^{4}=422481^{4}}$ 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

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[8] that if

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

where aibj are positive integers for all 1 ≤ in and 1 ≤ jm, then m + nk. In the special case m = 1, the conjecture states that if

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

(under the conditions given above) then nk − 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 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]

${\displaystyle (3a^{2}+5ab-5b^{2})^{3}+(4a^{2}-4ab+6b^{2})^{3}+(5a^{2}-5ab-3b^{2})^{3}=(6a^{2}-4ab+4b^{2})^{3}}$

A cube as the sum of three cubes can also be parameterized in one of two ways:[9]

{\displaystyle {\begin{aligned}a^{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{aligned}}}

The number 2 100 0003 can be expressed as the sum of three cubes in nine different ways.[9]

k = 4

{\displaystyle {\begin{aligned}422481^{4}&=95800^{4}+217519^{4}+414560^{4}\\[4pt]353^{4}&=30^{4}+120^{4}+272^{4}+315^{4}\end{aligned}}} (R. Frye, 1988);[4] (R. Norrie, smallest, 1911).[8]

k = 5

{\displaystyle {\begin{aligned}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{aligned}}}

(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

${\displaystyle 568^{7}=127^{7}+258^{7}+266^{7}+413^{7}+430^{7}+439^{7}+525^{7}}$

(M. Dodrill, 1999).[14]

k = 8

${\displaystyle 1409^{8}=90^{8}+223^{8}+478^{8}+524^{8}+748^{8}+1088^{8}+1190^{8}+1324^{8}}$

(S. Chase, 2000).[15]

References

1. ^ Dunham, William, ed. (2007). The Genius of Euler: Reflections on His Life and Work. The MAA. p. 220. ISBN 978-0-88385-558-4.
2. ^ Titus, III, Piezas (2005). "Euler's Extended Conjecture".
3. ^ a b 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.
4. ^ a b 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.
5. ^
6. ^ Piezas III, Tito (2010). "Sums of Three Fourth Powers (Part 1)". A Collection of Algebraic Identities. Retrieved April 11, 2022.
7. ^ 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, S2CID 58501120
8. 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.
9. ^ a b c
10. ^ 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.
11. ^
12. ^
13. ^ Giovanni Resta and Jean-Charles Meyrignac (2002). The Smallest Solutions to the Diophantine Equation ${\displaystyle a^{6}+b^{6}+c^{6}+d^{6}+e^{6}=x^{6}+y^{6}}$, Mathematics of Computation, v. 72, p. 1054 (See further work section).
14. ^
15. ^