Carol Primes

Carol primes can be expressed in the form (2n−1)2 − 2.

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127. As of July 2007, the largest known Carol number that is also a prime is the Carol number for n = 253987, which has 152916 digits. It was found by Cletus Emmanuel in May 2007, the initial studier of Carol primes, using the programs MultiSieve and PrimeFormGW. It is the 40th Carol prime.

The First Few Carol Primes
7

47

223

3967