A Carol number is an integer of the form 4ⁿ − 2^{n + 1} − 1. An equivalent formula is (2ⁿ − 1)² − 2. The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527 (sequence A093112 in OEIS). In mathematics, −1 is the integer greater than negative two (−2) and less than 0. ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... 47 (forty-seven) is the natural number following 46 and preceding 48. ... 223 is the natural number between 222 and 224. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

For n > 2, the binary representation of the nth Carol number is n − 2 consecutive ones, a single zero in the middle, and n + 1 more consecutive ones, or to put it algebraically,

So, for example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the nth Mersenne number and the nth Carol number is 2^{n + 1}. This gives yet another equivalent expression for Carol numbers, (2²ⁿ − 1) − 2^{n + 1}. The difference between the nth Kynea number and the nth Carol number is the (n + 2)th power of two. In mathematics, a Mersenne number is a number that is one less than a power of two. ... A Kynéa number (pronounced: Ka-Nay), named after Kynéa R. Griffith, is an integer of the form . ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

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 A091516). As of 2007, the largest known Carol number that is also a prime is the Carol number for n = 253987[1]. It was found by Cletus Emmanuel in May 2007, using MultiSieve and PrimeForm. It is the 40th Carol prime. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

Carol numbers and the formula to reveal them were first discovered and studied by Cletus Emmanuel who named them after a friend, Carol G. Kirnon.[2]

External links

• Eric W. Weisstein, Near-Square Prime at MathWorld.
• Prime Database entry for Carol(226749)
• Prime Database entry for Carol(248949)
• Generalized Woodall Primes (and related forms)