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 Template:Mvar and Template:Mvar greater than 1, if the sum of Template:Mvar many Template:Mvarth powers of positive integers is itself a Template:Mvarth power, then Template:Mvar is greater than or equal to Template:Mvar:

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

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case Template:Math: if $a_1^k+a_2^k=b^k,$ then Template:Math.

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

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. The general solution of the equation $x_1^3+x_2^3=x_3^3+x_4^3$ is

$$\begin{array}{rl}{x}_1& =\lambda (1-(a-3b)(a^2+3b^2))\\ x_2& =\lambda ((a+3b)(a^2+3b^2)-1)\\ x_3& =\lambda ((a+3b)-(a^2+3b^2{)}^2)\\ x_4& =\lambda ((a^2+3b^2{)}^2-(a-3b))\end{array}$$

where Template:Mvar, Template:Mvar 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 Template:Math. 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: $$\begin{array}{rl}144^5& =27^5+84^5+110^5+133^5\\ 14132^5& =(-220{)}^5+5027^5+6237^5+14068^5\\ 85359^5& =55^5+3183^5+28969^5+85282^5\end{array}$$ (Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

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

A particular case of Elkies' solutions can be reduced to the identity $$(85v^2+484v-313{)}^4+(68v^2-586v+10{)}^4+(2u{)}^4=(357v^2-204v+363{)}^4,$$ where $$u^2=22030+28849v-56158v^2+36941v^3-31790v^4.$$ This is an elliptic curve with a rational point at Template:Math. From this initial rational point, one can compute an infinite collection of others. Substituting Template:Math 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 Template:Math 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

File:Plato number.svg

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

Template:Main article In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if

$\sum _{i=1}^n{a}_i^k=\sum _{j=1}^m{b}_j^k$,

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

$\sum _{i=1}^n{a}_i^k=b^k$

(under the conditions given above) then Template:Math.

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

See OEIS: A347773 for more data.

Template:Math

$$3^3+4^3+5^3=6^3$$ (Plato's number 216) This is the case Template:Math, Template:Math of Srinivasa Ramanujan's formula $$(3a^2+5ab-5b^2{)}^3+(4a^2-4ab+6b^2{)}^3+(5a^2-5ab-3b^2{)}^3=(6a^2-4ab+4b^2{)}^3$$

A cube as the sum of three cubes can also be parameterized in one of two ways: $$\begin{array}{rl}{a}^3(a^3+b^3{)}^3& =b^3(a^3+b^3{)}^3+a^3(a^3-2b^3{)}^3+b^3(2a^3-b^3{)}^3\\ a^3(a^3+2b^3{)}^3& =a^3(a^3-b^3{)}^3+b^3(a^3-b^3{)}^3+b^3(2a^3+b^3{)}^3.\end{array}$$ The number 2,100,000³ can be expressed as the sum of three positive cubes in nine different ways.

Template:Math

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

Template:Math

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

(Lander & Parkin, 1966); (Lander, Parkin, Selfridge, smallest, 1967); (Lander, Parkin, Selfridge, second smallest, 1967); (Sastry, 1934, third smallest).

Template:Math

It has been known since 2002 that there are no solutions for Template:Math whose final term is ≤ 730000.

Template:Math

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

(M. Dodrill, 1999).

Template:Math

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

(S. Chase, 2000).

See also

• Jacobi–Madden equation
• Prouhet–Tarry–Escott problem
• Beal conjecture
• Pythagorean quadruple
• Generalized taxicab number
• Sums of powers, a list of related conjectures and theorems

External links

• Tito Piezas III, A Collection of Algebraic Identities Template:Webarchive
• 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!