# Carol number

A Carol number is an integer of the form $\displaystyle{ 4^n - 2^{n + 1} - 1, }$ or equivalently, $\displaystyle{ (2^n - 1)^2 - 2. }$ The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527 (sequence A093112 in the OEIS).

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,

$\displaystyle{ \sum_{i \ne n + 2}^{2n} 2^{i - 1}. }$

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 $\displaystyle{ 2^{n + 1} }$. This gives yet another equivalent expression for Carol numbers, $\displaystyle{ (2^{2n} - 1) - 2^{n + 1} }$. The difference between the n-th Kynea number and the n-th Carol number is the (n + 2)th power of two.

## Primes and modular relations Unsolved problem in mathematics:Are there infinitely many Carol primes?(more unsolved problems in mathematics)

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 (these are listed in Sloane's ).

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.

(As of April 2020), 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.