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A '''Carol number''' is an [[Integer|integer]] of the form <math>4^n - 2^{n + 1} - 1,</math> or equivalently, <math>(2^n - 1)^2 - 2.</math> The first few Carol numbers are:  &minus;1, 7, [[47 (number)|47]], [[223 (number)|223]], 959, 3967, 16127, 65023, 261119, 1046527 {{OEIS|id=A093112}}.  

The numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.<ref>[http://primes.utm.edu/bios/page.php?id=374 Cletus Emmanuel] at [[Prime Pages]]</ref><ref>
[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584 Message to Yahoo primenumbers group] from Cletus Emmanuel</ref>

==Binary representation==
For ''n'' > 2, the binary representation of the ''n''-th Carol number is ''n'' &minus; 2 consecutive ones, a single zero in the middle, and ''n'' + 1 more consecutive ones, or to put it algebraically,

:<math>\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 2''n''-th [[Mersenne prime|Mersenne number]] and the ''n''-th Carol number is <math>2^{n + 1}</math>. This gives yet another equivalent expression for Carol numbers, <math>(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|power of two]].

==Primes and modular relations==
{{unsolved|mathematics|Are there infinitely many Carol primes?}}

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a [[Prime number|prime number]], its index ''n'' cannot be of the form 3''x'' + 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 {{OEIS2C|id=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|emirp]].<ref>[http://primes.utm.edu/curios/page.php/16127.html Prime Curios 16127] at [[Prime Pages]]</ref> The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.<ref>[http://primes.utm.edu/curios/page.php/16769023.html Prime Curios 16769023] at [[Prime Pages]]</ref>

{{As of|2020|4}}, the largest known prime Carol number has index ''n'' = 695631, which has 418812 digits.<ref>[http://primes.utm.edu/primes/page.php?id=121905 Entry for 695631st Carol number] at [[Prime Pages]]</ref><ref>[http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm Carol and Kynea Prime Search] by Mark Rodenkirch</ref> It was found by Mark Rodenkirch in July 2016 using the programs CKSieve and PrimeFormGW. <ref>[https://www.mersenneforum.org/showpost.php?p=438288&postcount=81 Carol / Kynea Primes]</ref> It is the 44th Carol prime.

==References==
{{reflist}}

==External links==
* {{MathWorld|title=Near-Square Prime|urlname=Near-SquarePrime}}
* [http://primes.utm.edu/primes/page.php?id=121905 Prime Database entry for Carol(695631)]
* [http://harvey563.tripod.com/Carol_Kynea.txt Carol and Kynea Primes]
* [http://www.noprimeleftbehind.net/Carol-Kynea-prime-search.htm Carol and Kynea Prime Search]
[[Category:Integer sequences]]
[[Category:Unsolved problems in number theory]]

{{Prime number classes|state=collapsed}}
{{Classes of natural numbers |state=collapsed}}

{{Sourceattribution|Carol number|1}}