|
In mathematics, a Cullen number is a natural number of the form (written ). Cullen numbers were first studied by Fr. James Cullen in 1905. Cullen numbers are special cases of Proth numbers. == Properties == In 1976 Christopher Hooley showed that the natural density of positive integers for which ''Cn'' is a prime is of the order ''o(x)'' for . In that sense, almost all Cullen numbers are composite. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers ''n'' · 2''n''+''a'' + ''b'' where ''a'' and ''b'' are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for ''n'' equal: : 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 . Still, it is conjectured that there are infinitely many Cullen primes. , the largest known Cullen prime is 6679881 × 26679881 + 1. It is a megaprime with 2,010,852 digits and was discovered by a PrimeGrid participant from Japan. A Cullen number ''Cn'' is divisible by ''p'' = 2''n'' − 1 if ''p'' is a prime number of the form 8''k'' - 3; furthermore, it follows from Fermat's little theorem that if ''p'' is an odd prime, then p divides ''C''''m''(''k'') for each ''m''(''k'') = (2''k'' − ''k'') (''p'' − 1) − ''k'' (for ''k'' > 0). It has also been shown that the prime number ''p'' divides ''C''(''p'' + 1) / 2 when the Jacobi symbol (2 | ''p'') is −1, and that ''p'' divides ''C''(3''p'' − 1) / 2 when the Jacobi symbol (2 | ''p'') is +1. It is unknown whether there exists a prime number ''p'' such that ''C''''p'' is also prime. 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「Cullen number」の詳細全文を読む スポンサード リンク
|