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| '''Euler's conjecture''' is a disproved [[conjecture]] in [[mathematics]] related to [[Fermat's last theorem]] which was proposed by [[Leonhard Euler]] in 1769. It states that for all [[integers]] ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powers of positive integers is itself a ''k''th power, then ''n'' is greater than or equal to ''k''. | | Hello, I'm Annabelle, a 17 year old from Grapevine, United States.<br>My hobbies include (but are not limited to) Bus spotting, Insect collecting and watching Two and a Half Men.<br><br>my site how to get free fifa 15 coins ([http://intlfa.com/index.php?do=/blog/7362/fifa-15-coin-hack/add-comment/ browse around here]) |
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| In symbols, if
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| <math> | |
| \sum_{i=1}^{n} a_i^k = b^k
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| </math>
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| where <math>n>1</math> and <math>a_1, a_2, \dots, a_n, b</math> are positive integers, then <math>n\geq k</math>.
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| The conjecture represents an attempt to generalize [[Fermat's last theorem]], which could be seen as the special case of ''n'' = 2: if <math>a_1^k + a_2^k = b^k</math>, then <math>2 \geq k</math>.
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| Although the conjecture holds for the case of ''k'' = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for ''k'' = 4 and ''k'' = 5. It still remains unknown if the conjecture fails or holds for any value ''k'' ≥ 6.
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| == Background ==
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| Euler had an equality for four fourth powers <math>59^4 + 158^4 = 133^4 + 134^4</math>, this however is not a counterexample. He also provided a complete solution to the four cubes problem as in [[Plato's number]] <math>3^3+4^3+5^3=6^3</math> or the [[taxicab number]] 1729.<ref>{{cite book |title=The Genius of Euler: Reflections on His Life and Work |url=http://books.google.com/books?id=M4-zUnrSxNoC&pg=PA220 |editor=William Dunham |publisher= The MAA|page=220 |year=2007 |isbn=978-0-88385-558-4}}</ref><ref>{{cite web| url=http://www.oocities.org/titus_piezas/Equalsums.htm |title=Euler's Extended Conjecture |author=Titus Piezas III |year=2005}}</ref> The general solution for:
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| :<math>x_1^3+x_2^3=x_3^3+x_4^3</math>
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| is
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| :<math>x_1 = 1-(a-3b)(a^2+3b^2), x_2 = (a+3b)(a^2+3b^2)-1</math>
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| :<math>x_3 = (a+3b)-(a^2+3b^2)^2, x_4 = (a^2+3b^2)^2-(a-3b)</math>
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| where <math>a</math> and <math>b</math> are any integers.
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| == Counterexamples ==
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| === ''k'' = 5 ===
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| The conjecture was disproven by [[Leon J. Lander|L. J. Lander]] and [[Thomas Parkin|T. R. Parkin]] in [[1966 in science|1966]] when, through a direct computer search on a [[CDC 6600]], they found the following counterexample for ''k'' = 5:<ref>{{cite journal |author=L. J. Lander, T. R. Parkin |title=Counterexample to Euler's conjecture on sums of like powers |journal=Bull. Amer. Math. Soc. |volume=72 |year=1966 |pages=1079 |doi=10.1090/S0002-9904-1966-11654-3}}</ref>
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| ::27<sup>5</sup> + 84<sup>5</sup> + 110<sup>5</sup> + 133<sup>5</sup> = [[144 (number)|144]]<sup>5</sup>.
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| Yet another counterexample 85282<sup>5</sup> + 28969<sup>5</sup> + 3183<sup>5</sup> + 55<sup>5</sup> = 85359<sup>5</sup> was found by [[Jim Frye]] in 2004.
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| === ''k'' = 4 ===
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| In 1986, [[Noam Elkies]] found a method to construct an infinite series of counterexamples for the ''k'' = 4 case.<ref>{{cite journal |author=Noam Elkies |title=On ''A''<sup>4</sup> + ''B''<sup>4</sup> + ''C''<sup>4</sup> = ''D''<sup>4</sup> |journal=[[Mathematics of Computation]] |year=1988 |volume=51 |issue=184 |pages=825–835 |doi=10.2307/2008781 |mr=0930224 |jstor=2008781}}</ref> His smallest counterexample was the following:
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| ::2682440<sup>4</sup> + 15365639<sup>4</sup> + 18796760<sup>4</sup> = 20615673<sup>4</sup>.
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| A particular case of Elkies' solutions can be reduced to the identity:<ref>{{cite web|url=http://groups.google.com/group/sci.math/browse_thread/thread/15beef75eaddcb1b?hl=en#|title=Elkies' a^4+b^4+c^4 = d^4}}</ref><ref>{{cite web|url=http://sites.google.com/site/tpiezas/014|title=Sums of Three Fourth Powers}}</ref>
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| ::(85''v''<sup>2</sup>+484''v''−313)<sup>4</sup> + (68''v''<sup>2</sup>−586''v''+10)<sup>4</sup> + (2''u'')<sup>4</sup> = (357''v''<sup>2</sup>−204''v''+363)<sup>4</sup>
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| where
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| ::''u''<sup>2</sup> = 22030+28849''v''−56158''v''<sup>2</sup>+36941''v''<sup>3</sup>−31790''v''<sup>4</sup>.
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| This is an [[elliptic curve]] with a [[rational point]] with ''v''<sub>1</sub> = −31/467. From this initial rational point, one can further compute an infinite number of ''v<sub>i</sub>''. Substituting ''v''<sub>1</sub> into the identity and removing common factors gives the numerical example cited above.
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| In 1988, [[Roger Frye]] subsequently found the smallest possible counterexample for ''k'' = 4 by a direct computer search using techniques{{Citation needed|date=May 2009}} suggested by Elkies:
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| ::95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup>.
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| Moreover, this solution is the only one with values of the variables below 1,000,000.
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| == Generalizations ==
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| In 1967, L. J. Lander, T. R. Parkin, and [[John Selfridge]] conjectured<ref>{{cite journal |author=L. J. Lander, T. R. Parkin, J. L. Selfridge |title=A Survey of Equal Sums of Like Powers |journal=[[Mathematics of Computation]] |volume=21 |issue=99 |year=1967 |pages=446–459 |doi=10.1090/S0025-5718-1967-0222008-0 |jstor=2003249}}</ref> that if <math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k</math>, where ''a<sub>i</sub>'' ≠ ''b<sub>j</sub>'' are positive integers for all 1 ≤ ''i'' ≤ ''n'' and 1 ≤ ''j'' ≤ ''m'', then ''m''+''n'' ≥ ''k''.
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| This would imply as a special case that if
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| ::<math>\sum_{i=1}^{n} a_i^k = b^k</math>
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| (under the conditions given above) then ''n'' ≥ ''k''−1.
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| ==See also==
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| *[[Lander, Parkin, and Selfridge conjecture]]
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| * [[Jacobi–Madden equation]]
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| *[[Prouhet–Tarry–Escott problem]]
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| *[[Beal's conjecture]]
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| *[[Pythagorean quadruple]]
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| *[[Sums of powers]], a list of related conjectures and theorems
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| == References ==
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| {{reflist}}
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| == External links ==
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| * Tito Piezas III: [http://sites.google.com/site/tpiezas/Home/ A Collection of Algebraic Identities]
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| * [http://euler.free.fr/ EulerNet: Computing Minimal Equal Sums Of Like Powers]
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| * Jaroslaw Wroblewski [http://www.math.uni.wroc.pl/~jwr/eslp/ Equal Sums of Like Powers]
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| * {{MathWorld |title=Euler's Sum of Powers Conjecture |urlname=EulersSumofPowersConjecture}}
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| * {{MathWorld |title=Euler Quartic Conjecture |urlname=EulerQuarticConjecture}}
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| * {{MathWorld |title=Diophantine Equation--4th Powers |urlname=DiophantineEquation4thPowers}}
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| * [http://library.thinkquest.org/28049/Euler's%20conjecture.html Euler's Conjecture] at library.thinkquest.org
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| * [http://www.mathsisgoodforyou.com/conjecturestheorems/eulerconjecture.htm A simple explanation of Euler's Conjecture] at Maths Is Good For You!
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| * [http://www.sciencedaily.com/releases/2008/03/080314145039.htm Mathematicians Find New Solutions To An Ancient Puzzle]
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| * Ed Pegg Jr. [http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html Power Sums], Math Games
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| [[Category:Number theory]]
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| [[Category:Diophantine equations]]
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| [[Category:Disproved conjectures]]
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Hello, I'm Annabelle, a 17 year old from Grapevine, United States.
My hobbies include (but are not limited to) Bus spotting, Insect collecting and watching Two and a Half Men.
my site how to get free fifa 15 coins (browse around here)