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Several proofs for Fermat's Last Theorem for specific exponents have been developed. ==Mathematical preliminaries== Fermat's Last Theorem states that no three positive integers (''a'', ''b'', ''c'') can satisfy the equation ''a''''n'' + ''b''''n'' = ''c''''n'' for any integer value of ''n'' greater than two. If ''n'' equals two, the equation has infinitely many solutions, the Pythagorean triples. A solution (''a'', ''b'', ''c'') for a given ''n'' is equivalent to a solution for all the factors of ''n''. For illustration, let ''n'' be factored into ''g'' and ''h'', ''n'' = ''gh''. Then (''a''''g'', ''b''''g'', ''c''''g'') is a solution for the exponent ''h'' : (''a''''g'')''h'' + (''b''''g'')''h'' = (''c''''g'')''h'' Conversely, to prove that Fermat's equation has ''no'' solutions for ''n'' > 2, it suffices to prove that it has no solutions for ''n'' = 4 and for all odd primes ''p''. For any such odd exponent ''p'', every positive-integer solution of the equation ''a''''p'' + ''b''''p'' = ''c''''p'' corresponds to a general integer solution to the equation ''a''''p'' + ''b''''p'' + ''c''''p'' = 0. For example, if (3, 5, 8) solves the first equation, then (3, 5, −8) solves the second. Conversely, any solution of the second equation corresponds to a solution to the first. The second equation is sometimes useful because it makes the symmetry between the three variables ''a'', ''b'' and ''c'' more apparent. 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「Proof of Fermat's Last Theorem for specific exponents」の詳細全文を読む スポンサード リンク
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