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 $k$ th powers of positive integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ :

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

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

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

where $a$ , $b$ and ${\lambda }$ 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: ${\begin{aligned}144^{5}&=27^{5}+84^{5}+110^{5}+133^{5}\\14132^{5}&=(-220)^{5}+5027^{5}+6237^{5}+14068^{5}\\85359^{5}&=55^{5}+3183^{5}+28969^{5}+85282^{5}\end{aligned}}$ (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 $20615673^{4}=2682440^{4}+15365639^{4}+18796760^{4}.$

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 $v 1 = - ⁠ 31 / 467 ⁠$ . 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 $95800^{4}+217519^{4}+414560^{4}=422481^{4}$ 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

\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.

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]

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

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

${\begin{aligned}a^{3}(a^{3}+b^{3})^{3}&=b^{3}(a^{3}+b^{3})^{3}+a^{3}(a^{3}-2b^{3})^{3}+b^{3}(2a^{3}-b^{3})^{3}\\[6pt]a^{3}(a^{3}+2b^{3})^{3}&=a^{3}(a^{3}-b^{3})^{3}+b^{3}(a^{3}-b^{3})^{3}+b^{3}(2a^{3}+b^{3})^{3}\end{aligned}}$

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

$k = 4$

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

$k = 5$

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

(Lander & Parkin, 1966);^[10]^[11]^[12] (Lander, Parkin, Selfridge, smallest, 1967);^[8] (Lander, Parkin, Selfridge, second smallest, 1967);^[8] (Sastry, 1934, third smallest).^[8]

$k = 6$

It has been known since 2002 that there are no solutions for $k = 6$ whose final term is ≤ 730000.^[13]

$k = 7$

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

(M. Dodrill, 1999).^[14]

$k = 8$

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

(S. Chase, 2000).^[15]

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".
^ ^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.
^ ^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.
^ "Elkies' a⁴+b⁴+c⁴ = d⁴".
^ Piezas III, Tito (2010). "Sums of Three Fourth Powers (Part 1)". A Collection of Algebraic Identities. Retrieved April 11, 2022.
^ 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
^ ^a ^b ^c ^d ^e 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.
^ ^a ^b ^c "MathWorld : Diophantine Equation--3rd Powers".
^ 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.
^ "MathWorld: Diophantine Equation--5th Powers".
^ "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 $a^{6}+b^{6}+c^{6}+d^{6}+e^{6}=x^{6}+y^{6}$ , Mathematics of Computation, v. 72, p. 1054 (See further work section).
^ "MathWorld: Diophantine Equation--7th Powers".
^ "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!

[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".

[LanderParkin-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.

[Elkies1988-4] 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

[autogenerated446-8] 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.

[Weisstein-9] "MathWorld : Diophantine Equation--3rd Powers".

[mathologer-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 $a^{6}+b^{6}+c^{6}+d^{6}+e^{6}=x^{6}+y^{6}$ , Mathematics of Computation, v. 72, p. 1054 (See further work section).

[14] "MathWorld: Diophantine Equation--7th Powers".

[15] "MathWorld: Diophantine Equation--8th Powers".

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

Fahrer / Team für die Suche eingeben: