Cullen number

Cullen a number is a number of the form. With figures like these, the Reverend James Cullen in 1905 employed. It occurred to him that, that are not primes except C1 = 3, all numbers of this form to C99 composite numbers. His uncertainty was C53 respect of Allan JC Cunningham to 1906 dispelled by this divider was 5591. Cunningham showed that all Cn are assembled to n = 200, with a possible exception for n = 141

1958 confirmed Raphael M. Robinson, that C141 is a prime number, and showed that, with the exception of and all Cullen numbers of up composite numbers.

Wilfrid Keller has shown in 1984, the C4713, C5795, C6611 and C18496 are also prime numbers, but all other Cn with n ≤ 30000 composite Cullen numbers are.

Meanwhile, (June 2011) it is known that Cn for n prime numbers are the following: 32292, 32469, 59656, 90825, 262 419, 361 275, 481 899, 1354828, 6328548 and 6679881 Besides these, there are no Cullen primes up to n = 8771000. .

It is believed that there are infinitely many Cullen primes.

Woodall - number

A number of the form is called Cullen number of the second kind or Woodall 's number ( after HJ Woodall, who described it in 1917 ).

In the range of n ≤ 20000 are only the Woodall numbers C'2, C'3, C'6, C'30, C'75, C'81, C'115, C'123, C'249, C ' 362, C'384, C'462, C'512, C'751, C'882, C'5312, C'7755, C'9531, C'12379, C'15822 and C'18885 primes.

More Woodall primes are C'n for the following n: 22971, 23005, 98726, 143018, 151023, 667 071, 1195203, 1268979, 2013992, 2367906 and 3752948 found by PrimeGrid BOINC project. Besides these, there are no Woodall primes up to n = 8,866,000th

It is believed that there are infinitely many primes Woodall.

Generalized Cullen and Woodall numbers

Numbers of the form n · b n 1 is called a generalized Cullen numbers. Numbers of the form n · b n - 1 is called a generalized Woodall numbers