Carol number

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A Carol number, named after Carol G. Kirnon, is an integer of the form [4^n - 2^ - 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 in OEIS).

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,

[\sum_^ 2^]

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^]. This gives yet another equivalent expression for Carol numbers, [(2^ - 1) - 2^]. The difference between the nth Kynea number and the nth 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 ). As of 2005, the largest known Carol number that is also a prime is the Carol number for n = 226749, approximately 3.16937497264887779 × 10¹³⁶⁵¹⁶. It was found by Steven Harvey in early 2005, using MultiSieve and PrimeForm. It is the 38th Carol prime. These numbers were first encountered by Cletus Emmanuel in 1994, who consequently named them.

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