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

Carol numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.[1][2]

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,

\( \sum_{i \ne n + 2}^{2n} 2^{i - 1}. \)

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

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). As of July 2007, the largest known Carol number that is also a prime is the Carol number for n = 253987, which has 152916 digits.[3][4] It was found by Cletus Emmanuel in May 2007, using the programs MultiSieve and PrimeFormGW. It is the 40th Carol prime.

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.[5] The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.[6]
References

^ Cletus Emmanuel at Prime Pages
^ Message to Yahoo primenumbers group from Cletus Emmanuel
^ Entry for 253987th Carol number at Prime Pages
^ Carol Primes and Kynea Primes by Steven Harvey
^ Prime Curios 16127 at Prime Pages
^ Prime Curios 16769023 at Prime Pages

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