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:

<math display=block>a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k</math>

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case Template:Math: if <math>a_1^k + a_2^k = b^k,</math> 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 Template:Nowrap 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 Template:Nowrap or the taxicab number 1729.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The general solution of the equation <math>x_1^3+x_2^3=x_3^3+x_4^3</math> is

<math display=block>\begin{align}

 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{align}</math>

where Template:Mvar, Template:Mvar and <math>{\lambda}</math> 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.<ref name="LanderParkin">Template:Cite journal</ref> This was published in a paper comprising just two sentences.<ref name="LanderParkin" /> A total of three primitive (that is, in which the summands do not all have a common factor) counterexamples are known: <math display=block>\begin{align}

 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{align}</math> (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.<ref name="Elkies1988">Template:Cite journal</ref> His smallest counterexample was <math display=block>20615673^4 = 2682440^4 + 15365639^4 + 18796760^4.</math>

A particular case of Elkies' solutions can be reduced to the identity<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref> <math display=block>(85v^2 + 484v - 313)^4 + (68v^2 - 586v + 10)^4 + (2u)^4 = (357v^2 - 204v + 363)^4,</math> where <math display=block>u^2 = 22030 + 28849v - 56158v^2 + 36941v^3 - 31790v^4.</math> 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 <math display=block>95800^4 + 217519^4 + 414560^4 = 422481^4</math> 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.<ref>Template:Citation</ref>

Generalizations

File:Plato number.svg

One interpretation of Plato's number, Template:Nowrap

Template:Main article In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured<ref name="autogenerated446">Template:Cite journal</ref> that if

<math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k</math>,

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

<math>\sum_{i=1}^{n} a_i^k = b^k</math>

(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 Template:OEIS2C for more data.

Template:Math

<math display="block">3^3 + 4^3 + 5^3 = 6^3</math> (Plato's number 216) This is the case Template:Math, Template:Math of Srinivasa Ramanujan's formula<ref name=Weisstein>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display=block>(3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3</math>

A cube as the sum of three cubes can also be parameterized in one of two ways:<ref name=Weisstein/> <math display=block>\begin{align} 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{align}</math> The number 2,100,000³ can be expressed as the sum of three positive cubes in nine different ways.<ref name=Weisstein/>

Template:Math

<math display=block>\begin{align}

 422481^4 &= 95800^4 + 217519^4 + 414560^4 \\[4pt]
 353^4 &= 30^4 + 120^4 + 272^4 + 315^4

\end{align}</math> (R. Frye, 1988);<ref name="Elkies1988"/> (R. Norrie, smallest, 1911).<ref name="autogenerated446"/>

Template:Math

<math display=block>\begin{align}

 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{align}</math>

(Lander & Parkin, 1966);<ref name="mathologer">{{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> (Lander, Parkin, Selfridge, smallest, 1967);<ref name="autogenerated446"/> (Lander, Parkin, Selfridge, second smallest, 1967);<ref name="autogenerated446"/> (Sastry, 1934, third smallest).<ref name="autogenerated446"/>

Template:Math

It has been known since 2002 that there are no solutions for Template:Math whose final term is ≤ 730000.<ref>Giovanni Resta and Jean-Charles Meyrignac (2002). The Smallest Solutions to the Diophantine Equation <math>a^6+b^6+c^6+d^6+e^6=x^6+y^6</math>, Mathematics of Computation, v. 72, p. 1054 (See further work section).</ref>

Template:Math

<math display=block>568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7</math>

(M. Dodrill, 1999).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Template:Math

<math display=block>1409^8 = 90^8 + 223^8 + 478^8 + 524^8 + 748^8 + 1088^8 + 1190^8 + 1324^8 </math>

(S. Chase, 2000).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:EulersSumofPowersConjecture%7CEulersSumofPowersConjecture.html}} |title = Euler's Sum of Powers Conjecture |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:EulerQuarticConjecture%7CEulerQuarticConjecture.html}} |title = Euler Quartic Conjecture |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:DiophantineEquation4thPowers%7CDiophantineEquation4thPowers.html}} |title = Diophantine Equation--4th Powers |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Euler's Conjecture at library.thinkquest.org
• A simple explanation of Euler's Conjecture at Maths Is Good For You!