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In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation : satisfying ''x'' > ''y'' > 1 and ''n'', ''m'' > 2 are * (''x'', ''y'', ''m'', ''n'') = (5, 2, 3, 5); and * (''x'', ''y'', ''m'', ''n'') = (90, 2, 3, 13). This may be expressed as saying that 31 and 8191 are the only two numbers that are repunits with at least 3 digits in two different bases. Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions to the equations in (''x'',''y'',''m'',''n'') with prime divisors of ''x'' and ''y'' lying in a given finite set and that they may be effectively computed. ==See also== *Feit–Thompson conjecture *Nagell–Ljunggren equation 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「Goormaghtigh conjecture」の詳細全文を読む スポンサード リンク
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