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In mathematics, primorial primes are prime numbers of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (that is, the product of the first ''n'' primes). According to this definition, : ''pn''# − 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, ... : ''pn''# + 1 is prime for ''n'' = 1, 2, 3, 4, 5, 11, ... The first few primorial primes are :2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 , the largest known primorial prime is 1098133# − 1 (''n'' = 85586) with 476,311 digits, found by the PrimeGrid project.〔(Primegrid.com ); forum announcement, 2 March 2011〕 Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: 〔Michael Hardy and Catherine Woodgold, "Prime Simplicity", ''Mathematical Intelligencer'', volume 31, number 4, fall 2009, pages 44–52.〕 : Assume that the first ''n'' consecutive primes including 2 are the only primes that exist. If either ''pn''# + 1 or ''pn''# − 1 is a primorial prime, it means that there are larger primes than the ''n''th prime (if neither is a prime, that also proves the infinitude of primes, but less directly; note that each of these two numbers has a remainder of either ''p'' − 1 or 1 when divided by any of the first ''n'' primes, and hence cannot be a multiple of any of them). == See also == * Euclid number * Factorial prime * PrimeGrid * Primorial 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「Primorial prime」の詳細全文を読む スポンサード リンク
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