Английская Википедия: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 Шаблон:Mvar and Шаблон:Mvar greater than 1, if the sum of Шаблон:Mvar many Шаблон:Mvarth powers of positive integers is itself a Шаблон:Mvarth power, then Шаблон:Mvar is greater than or equal to Шаблон: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 Шаблон:Math: if <math>a_1^k + a_2^k = b^k,</math> then Шаблон:Math.
Although the conjecture holds for the case Шаблон:Math (which follows from Fermat's Last Theorem for the third powers), it was disproved for Шаблон:Math and Шаблон:Math. It is unknown whether the conjecture fails or holds for any value Шаблон:Math.
Background
Euler was aware of the equality Шаблон: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 Шаблон:Nowrap or the taxicab number 1729.[1][2] 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 &= 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{align}</math>
where Шаблон:Mvar and Шаблон:Mvar 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 Шаблон:Math.[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: <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 Шаблон:Math case.[4] 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[5][6] <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 Шаблон:Math. From this initial rational point, one can compute an infinite collection of others. Substituting Шаблон: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 Шаблон: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.[7]
Generalizations
Шаблон:Main article In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured[8] that if
- <math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k</math>,
where Шаблон:Math are positive integers for all Шаблон:Math and Шаблон:Math, then Шаблон:Math. In the special case Шаблон:Math, the conjecture states that if
- <math>\sum_{i=1}^{n} a_i^k = b^k</math>
(under the conditions given above) then Шаблон:Math.
The special case may be described as the problem of giving a partition of a perfect power into few like powers. For Шаблон:Math and Шаблон:Math or Шаблон:Math, there are many known solutions. Some of these are listed below.
See Шаблон:OEIS2C for more data.
Шаблон:Math
- Шаблон:Nowrap (Plato's number 216)
- This is the case Шаблон:Math, Шаблон:Math of Srinivasa Ramanujan's formula[9]
<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:[9]
<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 cubes in nine different ways.[9]
Шаблон: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);[4] (R. Norrie, smallest, 1911).[8]
Шаблон: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);[10][11][12] (Lander, Parkin, Selfridge, smallest, 1967);[8] (Lander, Parkin, Selfridge, second smallest, 1967);[8] (Sastry, 1934, third smallest).[8]
Шаблон:Math
As of 2002, there are no solutions for Шаблон:Math whose final term is ≤ 730000.[13]
Шаблон:Math
<math display=block>568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7</math>
(M. Dodrill, 1999).[14]
Шаблон: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).[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
External links
- Tito Piezas III, A Collection of Algebraic Identities
- 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
- Шаблон:MathWorld
- Шаблон:MathWorld
- Шаблон:MathWorld
- Euler's Conjecture at library.thinkquest.org
- A simple explanation of Euler's Conjecture at Maths Is Good For You!
Шаблон:Disproved conjectures Шаблон:Leonhard Euler
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ 3,0 3,1 Шаблон:Cite journal
- ↑ 4,0 4,1 Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Citation
- ↑ 8,0 8,1 8,2 8,3 8,4 Шаблон:Cite journal
- ↑ 9,0 9,1 9,2 Шаблон:Cite web
- ↑ Шаблон:Cite webШаблон:Cbignore
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ 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).
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
