Innumber theory,Euler's conjecture is adisprovedconjecture related toFermat's Last Theorem. It was proposed byLeonhard Euler in 1769. It states that for allintegersn andk greater than 1, if the sum ofn manykth powers of positive integers is itself akth power, thenn is greater than or equal tok:

$${\displaystyle a_1^k+a_2^k+\cdots +a_n^k=b^k⟹n\ge k}$$

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special casen = 2: if${\displaystyle a_1^k+a_2^k=b^k,}$ then2 ≥k.

Although the conjecture holds for the casek = 3 (which follows from Fermat's Last Theorem for the third powers), it was disproved fork = 4 andk = 5. It is unknown whether the conjecture fails or holds for any valuek ≥ 6.

Euler was aware of the equality59⁴ + 158⁴ = 133⁴ + 134⁴ involving sums of four fourth powers; this, however, is not acounterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as inPlato's number3³ + 4³ + 5³ = 6³ or thetaxicab number 1729.^[1][2] The general solution of the equation${\displaystyle x_1^3+x_2^3=x_3^3+x_4^3}$is

wherea,b and${\displaystyle \lambda }$ are any rational numbers.

Euler's conjecture was disproven byL. J. Lander andT. R. Parkin in 1966 when, through a direct computer search on aCDC 6600, they found a counterexample fork = 5.^[3] This was published in a paper comprising just two sentences.^[3] A total of three primitive (that is, in which thesummands do not all have acommon factor) counterexamples are known:(Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004).

In 1988,Noam Elkies published a method to construct an infinite sequence of counterexamples for thek = 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 anelliptic curve with arational point atv₁ = −⁠31/467⁠. From this initial rational point, one can compute an infinite collection of others. Substitutingv₁ into the identity and removing common factors gives the numerical example cited above.

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

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

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

wherea_i ≠b_j are positive integers for all1 ≤i ≤n and1 ≤j ≤m, thenm +n ≥k. In the special casem = 1, the conjecture states that if

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

(under the conditions given above) thenn ≥k − 1.

The special case may be described as the problem of giving apartition of a perfect power into few like powers. Fork = 4, 5, 7, 8 andn =k ork − 1, there are many known solutions. Some of these are listed below.

SeeOEIS: A347773 for more data.

$${\displaystyle 3^3+4^3+5^3=6^3}$$ (Plato's number 216)This is the casea = 1,b = 0 ofSrinivasa 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]The number 2,100,000³ can be expressed as the sum of three positive cubes in nine different ways.^[9]

(R. Frye, 1988);^[4] (R. Norrie, smallest, 1911).^[8]

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

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

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

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

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

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

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

• Tito Piezas III,A Collection of Algebraic IdentitiesArchived 2011-10-01 at theWayback 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 equationa⁶ +b⁶ =c⁶ +d⁶ +e⁶ +f⁶ +g⁶ fora,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!

• Diophantine equations
• Disproved conjectures
• Leonhard Euler

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