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:''This article refers to Carmichael's theorem about Fibonacci numbers. Carmichael's theorem may also refer to the recursive definition of the Carmichael function.'' Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for ''n'' greater than 12, the ''n''th Fibonacci number F(''n'') has at least one prime divisor that does not divide any earlier Fibonacci number. The only exceptions for ''n'' up to 12 are: :F(1)=1 and F(2)=1, which have no prime divisors :F(6)=8 whose only prime divisor is 2 (which is F(3)) :F(12)=144 whose only prime divisors are 2 (which is F(3)) and 3 (which is F(4)) If a prime ''p'' is a divisor of F(''n'') that does not divide any F(''m'') with ''m'' < ''n'', then ''p'' is called a ''characteristic factor'' or a ''primitive prime divisor'' of F(''n''). The smallest primitive prime divisor of F(''n'') are :1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, ... Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one primitive prime divisor. The theorem can be generalized from Fibonacci numbers to other Lucas sequences. For example, if ''n'' > 1, then the ''n''th Pell number has at least one prime divisor that does not divide any earlier Pell number. The smallest primitive prime divisor of ''n''th Pell number are :1, 2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 137, 199, 37, 19, 45697, 23, 229, 1153, 1549, 79, 53, 113, 44560482149, 31, 61, 665857, 52734529, 103, 1800193921, 73, 593, 9369319, 389, 241, ... ==See also== * Zsigmondy's theorem 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「Carmichael's theorem」の詳細全文を読む スポンサード リンク
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