Carol number

In number theory , a carol number is an integer of the form , or, equivalent, a number of the form with . They were first examined by Cletus Emmanuel, who named them after a friend, Carol G. Kimon. ${\displaystyle (2^{n}-1)^{2}-2}$ ${\displaystyle 4^{n}-2^{n+1}-1}$ ${\displaystyle n\geq 1}$

Examples

• The first carol numbers are the following:

−1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023, 67092479, 268402687, 1073676287, 4294836223, 17179607039, 68718952447, 2748768915733, 1743167407, 2748768918380, 17367274 392 387 387 397 398 716768915733, 109950953059227 4190207, 16769023 1046527 , ... (Follow A093112 in OEIS )

• The first carol prime numbers are the following:

7, 47, 223, 3967, 16127, 1046527, 16769023, 1073676287, 68718952447, 274876858367, 4398042316799, 1125899839733759, 18014398241046527, 1298074214633706835075030044377087 ... (sequence A091516 in OEIS )

They are called Carol primes .

• The seventh Carol number is also the fifth Carol prime number and is also a prime number if you reverse its digits (i.e. ). ${\displaystyle 16127}$ ${\displaystyle 72161}$

Such numbers are called Carol Mirp numbers .

There are currently only two Carol Mirp numbers known:

16127, 16769023

• The largest known Carol prime is and has digits. It was found by Mark Rodenkirch on July 16, 2016 with the programs CKSieve and PrimeFormGW . It's the 44th Carol prime number. ${\displaystyle (2^{695631}-1)^{2}-2}$ ${\displaystyle 418812}$

properties

• Each carol number of the form with has a binary representation , which is long digits, starts with ones, has a single zero in the middle and ends with additional ones. In other words: ${\displaystyle (2^{n}-1)^{2}-2}$ ${\displaystyle n>2}$ ${\displaystyle 2n}$ ${\displaystyle n-2}$ ${\displaystyle n+1}$

${\displaystyle (2^{n}-1)^{2}-2=\sum _{i=1 \atop i\not =n+2}^{2n}2^{i-1}}$

Example:

${\displaystyle 223=(2^{4}-1)^{2}-2={\underline {1}}\cdot 2^{7}+{\underline {1}}\cdot 2^{6}+{\underline {0}}\cdot 2^{5}+{\underline {1}}\cdot 2^{4}+{\underline {1}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=11011111_{2}}$

• The difference between the -th Mersenne number (also ) and the -th Carol number is . ${\displaystyle 2n}$ ${\displaystyle 2^{2n}-1}$ ${\displaystyle n}$ ${\displaystyle 2^{n+1}}$

So one could define the Carol numbers differently, namely as . ${\displaystyle (2^{2n}-1)-2^{n+1}}$

• The difference between the -th Kynea number and the -th Carol number is . ${\displaystyle n}$ ${\displaystyle (2^{n}+1)^{2}-2}$ ${\displaystyle n}$ ${\displaystyle 2^{n+2}}$
• If you start counting with the carol number 7, every third carol number is a multiple of . ${\displaystyle 7}$

Example:

${\displaystyle 65023=(2^{8}-1)^{2}-2}$is the sixth carol number after and is actually a multiple of . ${\displaystyle 7}$ ${\displaystyle 65023=9289\cdot 7}$ ${\displaystyle 7}$

• A carol number with for cannot be a prime number. ${\displaystyle (2^{n}-1)^{2}-2}$ ${\displaystyle n=3k+2}$ ${\displaystyle k>0}$

(follows from the property directly above)

Generalizations

A generalized Carol number to the base b is a number of the form with and a base . ${\displaystyle (b^{n}-1)^{2}-2}$ ${\displaystyle n\geq 1}$ ${\displaystyle b\geq 2}$

properties

• A generalized base carol number can only be prime if is an even number . ${\displaystyle b\geq 4}$ ${\displaystyle b}$

(If there were an odd number, every power would also be odd. If you subtract , the number is even. The square of this number is also even, and if you subtract it , it is still even and therefore certainly not prime (for ) proved this and the next property.) ${\displaystyle b}$ ${\displaystyle b^{n}}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle b\geq 4}$

• A generalized carol number with an odd base is always an even number . ${\displaystyle b}$
• A generalized base carol number is also a generalized base carol number . ${\displaystyle b^{n}}$ ${\displaystyle b}$
• The smallest , so that is prime (base ), are the following (for ): ${\displaystyle n\geq 1}$ ${\displaystyle ((2k)^{n}-1)^{2}-2}$ ${\displaystyle b=2k}$ ${\displaystyle k=1,2,3,4,\ldots ,100}$

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, ...

Example:

For you can see the number in the 6th position of the above list . ${\displaystyle n=6}$ ${\displaystyle n=3}$

In fact is a prime number. ${\displaystyle ((2\cdot 3)^{6}-1)^{2}-2=2176689023\in \mathbb {P} }$

The following is a table that shows the smallest generalized Carol primes with a base : ${\displaystyle b}$

${\displaystyle b}$	shape	Powers , so that generalized Carol numbers with a base , i.e. the form, are prime ${\displaystyle n\geq 1}$ ${\displaystyle b}$ ${\displaystyle (b^{n}-1)^{2}-2}$	OEIS episode
${\displaystyle 2}$	${\displaystyle (2^{n}-1)^{2}-2}$	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, ...	(Follow A091515 in OEIS )
${\displaystyle 3}$	${\displaystyle (3^{n}-1)^{2}-2}$	1 (leads to the even prime number ; there are no more powers ) ${\displaystyle p=2}$ ${\displaystyle n}$
${\displaystyle 4}$	${\displaystyle (4^{n}-1)^{2}-2}$	1, 2, 3, 5, 6, 9, 66, 162, 179, 393, 3231, 19506, 50112, 92790, 326745, 344021, ...
${\displaystyle 6}$	${\displaystyle (6^{n}-1)^{2}-2}$	1, 2, 6, 7, 20, 47, 255, 274, 279, 308, 1162, 2128, 3791, 9028, 9629, 10029, 13202, 38660, 46631, 48257, 117991, ...	(Follow A100901 in OEIS )
${\displaystyle 8}$	${\displaystyle (8^{n}-1)^{2}-2}$	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, ...
${\displaystyle 10}$	${\displaystyle (10^{n}-1)^{2}-2}$	1, 8, 21, 123, 4299, 6128, 11760, 18884, 40293, ...	(Follow A0100903 in OEIS )
${\displaystyle 12}$	${\displaystyle (12^{n}-1)^{2}-2}$	3, 29, 51, 7824, 15456, 22614, 28312, 47014, ...
${\displaystyle 14}$	${\displaystyle (14^{n}-1)^{2}-2}$	1, 6, 13, 45, 74, 240, 553, 12348, 13659, 50603, ...	(Follow A0100905 in OEIS )
${\displaystyle 16}$	${\displaystyle (16^{n}-1)^{2}-2}$	1, 3, 33, 81, 9753, 25056, 46395, ...
${\displaystyle 18}$	${\displaystyle (18^{n}-1)^{2}-2}$	2, 8, 30, 98, 110, 185, 912, 2514, 4074, 10208, 15123, 19395, ...
${\displaystyle 20}$	${\displaystyle (20^{n}-1)^{2}-2}$	1, 2, 53, 183, 1281, 1300, 8041, 29936, 72820, ...
${\displaystyle 22}$	${\displaystyle (22^{n}-1)^{2}-2}$	1, 8, 35, 88, 503, 8643, 8743, 14475, ...	(Follow A0100907 in OEIS )
${\displaystyle 24}$	${\displaystyle (24^{n}-1)^{2}-2}$	2, 27, 92, 4950, 20047, ...
${\displaystyle 26}$	${\displaystyle (26^{n}-1)^{2}-2}$	159, 879, 4744, 5602, 74387, ...
${\displaystyle 28}$	${\displaystyle (28^{n}-1)^{2}-2}$	1, 22, 127, 165, 2520, 6492, 6577, 22960, 25528, ...
${\displaystyle 30}$	${\displaystyle (30^{n}-1)^{2}-2}$	1, 6, 19, 30, 166, 495, 769, 826, 1648, 3993, ...
${\displaystyle 32}$	${\displaystyle (32^{n}-1)^{2}-2}$	2, 3, 5, 11, 35, 63, 87, 37116, 130698, ...
${\displaystyle 34}$	${\displaystyle (34^{n}-1)^{2}-2}$	1, 4, 258, ...
${\displaystyle 36}$	${\displaystyle (36^{n}-1)^{2}-2}$	1, 3, 10, 137, 154, 581, 1064, 4514, 6601, 19330, ...
${\displaystyle 38}$	${\displaystyle (38^{n}-1)^{2}-2}$	1, 2, 13, 560, 28933, ...
${\displaystyle 40}$	${\displaystyle (40^{n}-1)^{2}-2}$	4, 15, 39, 138, 2153, 4084, 5639, ...
${\displaystyle 42}$	${\displaystyle (42^{n}-1)^{2}-2}$	3, 6, 14, 15, 29, 78, 195, 255, 272, 713, 2526, 4852, 10573, ...
${\displaystyle 44}$	${\displaystyle (44^{n}-1)^{2}-2}$	1, 7, 30, 90, 1288, 1947, 12909, 25786, ...
${\displaystyle 46}$	${\displaystyle (46^{n}-1)^{2}-2}$	12, 269, 1304, 5172, ...
${\displaystyle 48}$	${\displaystyle (48^{n}-1)^{2}-2}$	1, 2, 4, 6, 12, 13, 3882, 6123, 15067, 15085, ...
${\displaystyle 50}$	${\displaystyle (50^{n}-1)^{2}-2}$	1, 3, 4, 9, 31, 66, 115, 430, 1233, 2546, 2674, 6360, 53351, 69033, 69157, ...

The largest known generalized Carol prime is and has digits. She was found by Karsten Bonath on March 1, 2019. It is the third Kynea prime with this base. ${\displaystyle (290^{124116}-1)^{2}-2}$ ${\displaystyle 611246}$

Further generalizations

A positive integer of the form called Noddy number ( Noddy number ). ${\displaystyle (2^{n}-1)^{3}+2}$

The smallest prime Noddy numbers are the following:

0, 1, 2, 6, 10, 16, 48, 70, 1196, 3958, 57096, 59556, 62440, 70362, ... (sequence A0100899 in OEIS )

See also

• Kynea number

Web links

• Eric W. Weisstein : Near-Square Prime . In: MathWorld (English).
• Mark Rodenkirch, Gary Barnes, Karsten Bonath: Carol and Kynea Prime Search .
• Carol and Kynea primes

Individual evidence

1. ↑ Cletus Emmanuel on Prime Pages
2. ↑ Message to Yahoo prime numbers group Cletus Emmanuel
3. ↑ (2 ⁶⁹⁵⁶³¹ -1) ² -2 on The Lagest Known Primes!
4. ↑ ^{a b} Carol and Kynea Prime Search by Mark Rodenkirch, Gary Barnes and Karsten Bonath
5. ↑ (290 ^{124 116} -1) ² -2 on The Lagest Known Primes!
6. ↑ ^{a b} Carol and Kynea primes