Carol number

A Carol number is an integer of the form [math]\displaystyle{ 4^n - 2^{n + 1} - 1, }[/math] or equivalently, [math]\displaystyle{ (2^n - 1)^2 - 2. }[/math] 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,

[math]\displaystyle{ \sum_{i \ne n + 2}^{2n} 2^{i - 1}. }[/math]

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 [math]\displaystyle{ 2^{n + 1} }[/math]. This gives yet another equivalent expression for Carol numbers, [math]\displaystyle{ (2^{2n} - 1) - 2^{n + 1} }[/math]. 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), 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.

External links

• Weisstein, Eric W.. "Near-Square Prime". http://mathworld.wolfram.com/Near-SquarePrime.html. 
• Prime Database entry for Carol(695631)
• Carol and Kynea Primes
• Carol and Kynea Prime Search