Lander, Parkin, and Selfridge conjecture

Diophantine equations, such as the integer version of the equation that appears in the Pythagorean theorem, have been studied for their integer solution properties for centuries. Fermat's Last Theorem states that for powers greater than 2, the equation has no solutions in non-zero integers . Extending the number of terms on either or both sides, and allowing for higher powers than 2, led to Leonhard Euler to propose in 1769 that for all integers and greater than 1, if the sum of th powers of positive integers is itself a th power, then is greater than or equal to . In symbols, if \sum_{i=1}^{n} a_i^k = b^k where and a_1, a_2, \dots, a_n, b are positive integers, then his conjecture was that . In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for : :. In subsequent years, further counterexamples were found, including for . The latter disproved the more specific Euler quartic conjecture, namely that has no positive integer solutions. In fact, the smallest solution, found in 1988, is :. == Conjecture ==

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if \sum_{i=1}^n a_i^k = \sum_{j=1}^m b_j^k, where are positive integers for all and , then . The equal sum of like powers formula is often abbreviated as . Small examples with m=n=\tfrac{k}{2} (related to generalized taxicab number) include : (known to Euler) and : (found by K. Subba Rao in 1934). The conjecture implies in the special case of that if \sum_{i=1}^n a_i^k = b^k (under the conditions given above) then . is a solution for For this special case of , some of the known solutions satisfying the proposed constraint with , where terms are positive integers, hence giving a partition of a power into like powers, are: ; :: ; :: (Roger Frye, 1988) :: (R. Norrie, 1911) Fermat's Last Theorem implies that for the conjecture is true. ; :: (Lander, Parkin, 1966) :: (Sastry, 1934, third smallest) ; ::(None known. As of 2002, there are no solutions with . ) ; :: (M. Dodrill, 1999) ; :: (Scott Chase, 2000) ; ::(None known.) == Current status==

It is not known if the conjecture is true, or if nontrivial solutions exist that would be counterexamples, such as for . ==See also==