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In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes. ==Definition== A Cunningham chain of the first kind of length ''n'' is a sequence of prime numbers (''p''1, ..., ''p''''n'') such that for all 1 ≤ ''i'' < ''n'', ''p''''i''+1 = 2''p''''i'' + 1. (Hence each term of such a chain except the last one is a Sophie Germain prime, and each term except the first is a safe prime). It follows that : Or, by setting (the number is not part of the sequence and need not be a prime number), we have Similarly, a Cunningham chain of the second kind of length ''n'' is a sequence of prime numbers (''p''1,...,''p''''n'') such that for all 1 ≤ ''i'' < ''n'', ''p''''i''+1 = 2''p''''i'' − 1. It follows that the general term is : Now, by setting , we have . Cunningham chains are also sometimes generalized to sequences of prime numbers (''p''1, ..., ''p''''n'') such that for all 1 ≤ ''i'' ≤ ''n'', ''p''''i''+1 = ''ap''''i'' + ''b'' for fixed coprime integers ''a'', ''b''; the resulting chains are called generalized Cunningham chains. A Cunningham chain is called complete if it cannot be further extended, i.e., if the previous or next term in the chain would not be a prime number anymore. 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「cunningham chain」の詳細全文を読む スポンサード リンク
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