View source for Kynea number

A '''Kynea number''' is an integer of the form

:<math>4^n + 2^{n + 1} - 1</math>.

An equivalent formula is

:<math>(2^n + 1)^2 - 2</math>.

This indicates that a Kynea number is the ''n''th power of 4 plus the (''n'' + 1)th [[Mersenne number]]. 
Kynea numbers were studied by Cletus Emmanuel who named them after a baby girl.<ref>[https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/14584]</ref>

The sequence of Kynea numbers starts with:
: [[7 (number)|7]], [[23 (number)|23]], [[79 (number)|79]], 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... {{OEIS|id=A093069}}.

== Properties ==

The [[Binary numeral system|binary representation]] of the ''n''th Kynea number is a single leading one, followed by ''n'' - 1 consecutive zeroes, followed by ''n'' + 1 consecutive ones, or to put it algebraically:

:<math>4^n + \sum_{i = 0}^n 2^i.</math>

So, for example, 23 is 10111 in binary, 79 is 1001111, etc. The difference between the ''n''th Kynea number and the ''n''th [[Carol number]] is the (''n'' + 2)th [[power of two]].

== Prime Kynea numbers ==
{| class="infobox" style="width: 15em; font-size: 90%; text-align: center;"
|-----
| colspan="3" align="center" | '''Kynea numbers'''
|-----
| '''n''' || Decimal || Binary
|-----
| '''1''' || 7 || 111
|-----
| '''2''' || 23 || 10111
|-----
| '''3''' || 79 || 1001111
|-----
| '''4''' || 287 || 100011111
|-----
| '''5''' || 1087 || 10000111111
|-----
| '''6''' || 4223 || 1000001111111
|-----
| '''7''' || 16639 || 100000011111111
|-----
| '''8''' || 66047 || 10000000111111111
|-----
| '''9''' || 263167 || 1000000001111111111
|}
Starting with 7, every third Kynea number is a multiple of 7. Thus, for a Kynea number to be a [[prime number]], its index ''n'' cannot be of the form 3''x'' + 1 for ''x'' > 0. The first few Kynea numbers that are also prime are 7, 23, 79, 1087, 66047, 263167, 16785407 {{OEIS|id=A091514}}.

Their n values are: 1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, ... {{OEIS|id=A091513}}.

{{As of|2019|7}}, the largest known prime Kynea number has index ''n'' = 852770, which has 513419 digits.<ref>[http://primes.utm.edu/primes/page.php?id=126646 Entry for 852770th Kynea 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 Ryan Propper in July 2019 using the programs CKSieve and PrimeFormGW.  It is the 51st Kynea prime.


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==References==
<references/>

==External links==
* {{MathWorld|title=Near-Square Prime|urlname=Near-SquarePrime}}
* [http://primes.utm.edu/primes/page.php?id=121801 Prime Database entry for Kynea(661478)]
* [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]
* [https://www.rieselprime.de/ziki/Carol-Kynea_prime Carol-Kynea prime] in [[Prime wiki]]

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

{{DEFAULTSORT:Kynea Number}}
[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]