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 (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 OEIS: A091516).
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[update], 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.