Carol number

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,

${\displaystyle \sum _{i\neq n+2}^{2n}2^{i-1}.}$

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.

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.[3] The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.[4]

As of April 2020, the largest known prime Carol number has index n = 695631, which has 418812 digits.[5][6] It was found by Mark Rodenkirch in July 2016 using the programs CKSieve and PrimeFormGW. [7] It is the 44th Carol prime.

• Weisstein, Eric W. "Near-Square Prime". MathWorld.
• Prime Database entry for Carol(695631)
• Carol and Kynea Primes
• Carol and Kynea Prime Search