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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Euler's conjecture is a disproved
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
related to
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
. It was proposed by
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
in 1769. It states that for all
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
and greater than 1, if the sum of many th powers of positive integers is itself a th power, then is greater than or equal to : a_1^k + a_2^k + \dots + a_n^k = b^k \implies n \ge k The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case : if a_1^k + a_2^k = b^k, then . Although the conjecture holds for the case (which follows from Fermat's Last Theorem for the third powers), it was disproved for and . It is unknown whether the conjecture fails or holds for any value .


Background

Euler was aware of the equality involving sums of four fourth powers; this, however, is not a
counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
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 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 x_1 &=\lambda( 1-(a-3b)(a^2+3b^2)) \\ pt x_2 &=\lambda( (a+3b)(a^2+3b^2)-1 )\\ pt x_3 &=\lambda( (a+3b)-(a^2+3b^2)^2 )\\ pt x_4 &= \lambda( (a^2+3b^2)^2-(a-3b)) \end where , and 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 . 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 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 (Lander & Parkin, 1966); (Scher & Seidl, 1996); (Frye, 2004). In 1988, Noam Elkies published a method to construct an infinite sequence of counterexamples for the 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 In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
with a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
at . From this initial rational point, one can compute an infinite collection of others. Substituting 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 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

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if :\sum_^ a_i^k = \sum_^ b_j^k, where are positive integers for all and , then . In the special case , the conjecture states that if :\sum_^ a_i^k = b^k (under the conditions given above) then . The special case may be described as the problem of giving a partition of a perfect power into few like powers. For and or , there are many known solutions. Some of these are listed below. See for more data.


3^3 + 4^3 + 5^3 = 6^3 ( Plato's number 216) This is the case , of
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
'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 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 \\ pta^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 The number 2,100,0003 can be expressed as the sum of three positive cubes in nine different ways.


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


\begin 144^5 &= 27^5 + 84^5 + 110^5 + 133^5 \\ pt 72^5 &= 19^5 + 43^5 + 46^5 + 47^5 + 67^5 \\ pt 94^5 &= 21^5 + 23^5 + 37^5 + 79^5 + 84^5 \\ pt 107^5 &= 7^5 + 43^5 + 57^5 + 80^5 + 100^5 \end (Lander & Parkin, 1966); (Lander, Parkin, Selfridge, smallest, 1967); (Lander, Parkin, Selfridge, second smallest, 1967); (Sastry, 1934, third smallest).


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


568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 (M. Dodrill, 1999).


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


References


External links

* Tito Piezas III
A Collection of Algebraic Identities
* Jaroslaw Wroblewski
Equal Sums of Like Powers
* Ed Pegg Jr.

* 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 ''a''6 + ''b''6 = ''c''6 + ''d''6 + ''e''6 + ''f''6 + ''g''6 for ''a'',''b'',''c'',''d'',''e'',''f'',''g'' < 250000 found with a distributed Boinc project

EulerNet: Computing Minimal Equal Sums Of Like Powers
* * *

at library.thinkquest.org

at Maths Is Good For You! {{Leonhard Euler Diophantine equations Disproved conjectures Leonhard Euler