Carol number


A Carol number is an integer of the form or equivalently, The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527.
The numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.

Binary representation

For n > 2, the binary representation of the n-th Carol number is n − 2 consecutive ones, a single zero in the middle, and n + 1 more consecutive ones, or to put it algebraically,
For example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the 2n-th Mersenne number and the n-th Carol number is. This gives yet another equivalent expression for Carol numbers,. The difference between the n-th Kynea number and the n-th Carol number is the th power of two.

Primes and modular relations

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.
The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol emirp. The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.
, the largest known prime Carol number has index n = 695631, which has 418812 digits. It was found by Mark Rodenkirch in July 2016 using the programs CKSieve and PrimeFormGW. It is the 44th Carol prime.