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

$${\displaystyle {\begin{aligned}x_{1}&=1-(a-3b)(a^{2}+3b^{2})\\[2pt]x_{2}&=(a+3b)(a^{2}+3b^{2})-1\\[2pt]x_{3}&=(a+3b)-(a^{2}+3b^{2})^{2}\\[2pt]x_{4}&=(a^{2}+3b^{2})^{2}-(a-3b)\end{aligned}}}$$

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:

$${\displaystyle {\begin{aligned}144^{5}&=27^{5}+84^{5}+110^{5}+133^{5}\\14132^{5}&=(-220)^{5}+5027^{5}+6237^{5}+14068^{5}\\85359^{5}&=55^{5}+3183^{5}+28969^{5}+85282^{5}\end{aligned}}}$$

(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

$${\displaystyle 20615673^{4}=2682440^{4}[phone]^{4}[phone]^{4}.}$$

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

One interpretation of Plato's number, 3³ + 4³ + 5³ = 6³

Main article: Lander, Parkin, and Selfridge conjecture

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 a_i ≠ b_j 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

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

(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]

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

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]

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

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. 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. 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. "Elkies' a⁴+b⁴+c⁴ = d⁴".
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. "MathWorld : Diophantine Equation--3rd Powers".
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. "MathWorld: Diophantine Equation--5th Powers".
12. "A Table of Fifth Powers equal to Sums of Five Fifth Powers".
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. "MathWorld: Diophantine Equation--7th Powers".
15. "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!