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The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. == Background == Diophantine equations, such as the integer version of the equation ''a''2 + ''b''2 = ''c''2 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 ''a''''k'' + ''b''''k'' = ''c''''k'' has no solutions with three positive integers ''a'', ''b'', ''c''. 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 ''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''. In symbols, if where ''n'' > 1 and are positive integers, then his conjecture was that ''n'' ≥ ''k''. In 1966, a counterexample to Euler's sum of powers conjecture was found by L. J. Lander and T. R. Parkin for ''k'' = 5: ::275 + 845 + 1105 + 1335 = 1445. In subsequent years further counterexamples were found, including for ''k'' = 4. The latter disproved the more specific Euler quartic conjecture, namely that ''a''4 + ''b''4 + ''c''4 = ''d''4, has no positive integer solutions. In fact, the smallest solution, found in 1988, is ::4145604 + 2175194 + 958004 = 4224814. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lander, Parkin, and Selfridge conjecture」の詳細全文を読む スポンサード リンク
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