# Carol number

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

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

## 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,

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

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 ${\displaystyle 2^{n+1}}$. 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.

## Primes and modular relations

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 ).

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 February 2018, 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. It is the 44th Carol prime.

## Generalizations

A generalized Carol number base b is defined to be a number of the form (bn−1)2 − 2 with n ≥ 1, a generalized Carol number base b can be prime only if b is even, since if b is odd, then all generalized Carol numbers base b are even and thus not prime. A generalized Carol number to base bn is also a generalized Carol number to base b.

Least n ≥ 1 such that ((2b)n−1)2 − 2 is prime are

2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 159, 1, 1, 2, 1, 1, 1, 4, 3, 1, 12, 1, 1, 2, 9, 1, 88, 2, 1, 1, 12, 4, 1, 1, 183, 1, 1, 320, 24, 4, 3, 2, 1, 3, 1, 5, 2, 4, 2, 1, 2, 1, 705, 2, 3, 29, 1, 1, 1, 4836, 20, 1, 135, 1, 4, 1, 6, 1, 15, 3912, 1, 2, 8, 3, 24, 1, 14, 4, 1, 2, 321, 11, 1, 174, 1, 6, 1, 42, 310, 1, 2, 27, 2, 1, 29, 3, 103, 20, ...
 b numbers n ≥ 1 such that (bn−1)2 − 2 is prime (these n are checked up to 30000) OEIS sequence 2 2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637, 100224, 108127, 110953, 175749, 185580, 226749, 248949, 253987, 520363, 653490, 688042, 695631, ... A091515 4 1, 2, 3, 5, 6, 9, 66, 162, 179, 393, 3231, 19506, 50112, 92790, 326745, 344021, ... 6 1, 2, 6, 7, 20, 47, 255, 274, 279, 308, 1162, 2128, 3791, 9028, 9629, 10029, 13202, 38660, 46631, 48257, 117991, ... A100901 8 1, 2, 4, 5, 6, 7, 9, 43, 44, 53, 57, 105, 108, 131, 145, 262, 569, 2154, 4763, 13004, 33408, 58583, 61860, 75583, 82983, 217830, 231877, ... 10 1, 8, 21, 123, 4299, 6128, 11760, 18884, 40293, ... A100903 12 3, 29, 51, 7824, 15456, 22614, 28312, 47014, ... 14 1, 6, 13, 45, 74, 240, 553, 12348, 13659, 50603, ... A100905 16 1, 3, 33, 81, 9753, 25056, 46395, ... 18 2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395, ... 20 1, 2, 53, 183, 1281, 1300, 8041, 29936, 72820, ... 22 1, 8, 35, 88, 503, 8643, 8743, 14475, ... A100907 24 2, 27, 92, 4950, 20047, ... 26 159, 879, 4744, 5602, 74387, ... 28 1, 22, 127, 165, 2520, 6492, 6577, 22960, 25528, ... 30 1, 6, 19, 30, 166, 495, 769, 826, 1648, 3993, ... 32 2, 3, 5, 11, 35, 63, 87, 37116, 130698, ... 34 1, 4, 258, ... 36 1, 3, 10, 137, 154, 581, 1064, 4514, 6601, 19330, ... 38 1, 2, 13, 560, 28933, ... 40 4, 15, 39, 138, 2153, 4084, 5639, ... 42 3, 6, 14, 15, 29, 78, 195, 255, 272, 713, 2526, 4852, 10573, ... 44 1, 7, 30, 90, 1288, 1947, 12909, 25786, ... 46 12, 269, 1304, 5172, ... 48 1, 2, 4, 6, 12, 13, 3882, 6123, 15067, 15085, ... 50 1, 3, 4, 9, 31, 66, 115, 430, 1233, 2546, 2674, 6360, 53351, 69033, 69157, ...

As of March 2019, the largest known generalized Carol prime is (290124116−1)2 − 2[7].

## References

1. ^
2. ^ Message to Yahoo primenumbers group from Cletus Emmanuel
3. ^
4. ^
5. ^
6. ^ Carol and Kynea Prime Search by Mark Rodenkirch
7. ^