Kynea number

In number theory , a Kynea 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 baby, Kynéa R. Griffith. ${\displaystyle (2^{n}+1)^{2}-2}$ ${\displaystyle 2^{2n}+2^{n+1}-1}$ ${\displaystyle n\geq 1}$

Examples

• The first Kynea numbers are the following:

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247, 268468223, 1073807359, 4295098367, 17180131327, 68720001023, 274878955519, 7097539760, 274878955519, 70975014723, 274878955519, 70975014723, 274878955519. ... (Follow A093069 in OEIS )

• The first Kynea prime numbers are as follows

(This is not included in the list because of , but would also have the form ): ${\displaystyle 2}$ ${\displaystyle n\geq 1}$ ${\displaystyle (2^{n}+1)^{2}-2=(2^{0}+1)^{2}-2=4-2=2}$

7, 23, 79, 1087, 66047, 263167, 16785407, 1073807359, 17180131327, 68720001023, 4398050705407, 70368760954879, 18014398777917439, 18446744082299486207, 5070602400912922109586440191999 ... (sequence A091514 in OEIS )

They are called Kynea primes .

• The largest known Kynea prime is and has digits. It was found by Mark Rodenkirch in June 2016 with the programs CKSieve and PrimeFormGW . It's the 50th Kynea prime number. ${\displaystyle (2^{661478}+1)^{2}-2}$ ${\displaystyle 398250}$

properties

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

${\displaystyle (2^{n}+1)^{2}-2=4^{n}+\sum _{i=0}^{n}2^{i}}$

Example:

${\displaystyle 287=(2^{4}+1)^{2}-2={\underline {1}}\cdot 2^{8}+{\underline {0}}\cdot 2^{7}+{\underline {0}}\cdot 2^{6}+{\underline {0}}\cdot 2^{5}+{\underline {1}}\cdot 2^{4}+{\underline {1}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=100011111_{2}}$

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

Example:

${\displaystyle 16639=(2^{7}+1)^{2}-2}$is the sixth carol number after and is actually a multiple of . ${\displaystyle 7}$ ${\displaystyle 16639=2377\cdot 7}$ ${\displaystyle 7}$

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

(follows from the property directly above)

• A Kynea number is the sum of the -th power of 4 and the -th Mersenne number . ${\displaystyle (2^{n}+1)^{2}-2}$ ${\displaystyle n}$ ${\displaystyle (n+1)}$

Generalizations

A generalized base b Kynea number 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 Kynea number can only be prime if is an even number . ${\displaystyle b}$ ${\displaystyle b}$

(If there were an odd number, every power would also be odd. If you add to this, 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. So this and that next property proved.) ${\displaystyle b}$ ${\displaystyle b^{n}}$ ${\displaystyle 1}$ ${\displaystyle 2}$

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

1, 1, 1, 1, 22, 1, 1, 2, 1, 1, 3, 24, 1, 1, 2, 1, 1, 1, 6, 2, 1, 3, 1, 1, 4, 3, 1, 8, 2, 1, 1, 2, 172, 1, 1, 354, 1, 1, 3, 29, 3, 423, 8, 1, 11, 1, 5, 2, 4, 11, 1, 6, 1, 3, 57, 24, 368, 1, 1, 1, 11, 19, 1, 3, 1, 13, 1, 12, 1, 41, 3, 1, 3, 4, 4, 2, 1, 152, 1893, 1, 12, 6, 2, 1, 11, 1, 2, 1, 3, 14, 1, 2, 6, 2, 1, 1017, 3, 30, 6, 3, ...

Example:

For you can find the number at the 11th position in the above list . ${\displaystyle k=11}$ ${\displaystyle n=3}$

In fact is a prime number. ${\displaystyle ((2\cdot 11)^{3}+1)^{2}-2=113401199\in \mathbb {P} }$

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

${\displaystyle b}$	shape	Powers , so that generalized Kynea numbers with a base , i.e. the form, are prime ${\displaystyle n\geq 1}$ ${\displaystyle b}$ ${\displaystyle (b^{n}+1)^{2}-2}$	OEIS episode
${\displaystyle 2}$	${\displaystyle (2^{n}-1)^{2}-2}$	1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, 491, 501, 507, 555, 591, 680, 800, 1070, 1650, 2813, 3281, 4217, 5153, 6287, 6365, 10088, 10367, 37035, 45873, 69312, 102435, 106380, 108888, 110615, 281621, 369581, 376050, 442052, 621443, 661478, ...	(Follow A091513 in OEIS )
${\displaystyle 4}$	${\displaystyle (4^{n}-1)^{2}-2}$	1, 4, 6, 9, 16, 90, 121, 340, 400, 535, 825, 5044, 34656, 53190, 54444, 188025, 221026, 330739, ...
${\displaystyle 6}$	${\displaystyle (6^{n}-1)^{2}-2}$	1, 2, 3, 4, 9, 12, 30, 49, 56, 115, 118, 376, 432, 1045, 1310, 6529, 7768, 8430, 21942, 26930, 33568, 50800, ...	(Follow A100902 in OEIS )
${\displaystyle 8}$	${\displaystyle (8^{n}-1)^{2}-2}$	1, 3, 4, 5, 6, 7, 9, 17, 29, 60, 167, 169, 185, 197, 550, 12345, 15291, 23104, 34145, 35460, 36296, 125350, ...
${\displaystyle 10}$	${\displaystyle (10^{n}-1)^{2}-2}$	22, 351, 1061, ...	(Follow A100904 in OEIS )
${\displaystyle 12}$	${\displaystyle (12^{n}-1)^{2}-2}$	1, 2, 8, 60, 513, 1047, 7021, 7506, ...
${\displaystyle 14}$	${\displaystyle (14^{n}-1)^{2}-2}$	1, 5, 60, 72, 118, 181, 245, 310, 498, 820, 962, 2212, 3928, 5844, 5937, ...	(Follow A100906 in OEIS )
${\displaystyle 16}$	${\displaystyle (16^{n}-1)^{2}-2}$	2, 3, 8, 45, 170, 200, 2522, 17328, 26595, 27222, 110513, ...
${\displaystyle 18}$	${\displaystyle (18^{n}-1)^{2}-2}$	1, 10, 21, 25, 31, 1083, 40485, ...
${\displaystyle 20}$	${\displaystyle (20^{n}-1)^{2}-2}$	1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935, ...
${\displaystyle 22}$	${\displaystyle (22^{n}-1)^{2}-2}$	3, 166, 814, 1851, 2197, 3172, 3865, 19791, ...	(Follow A100908 in OEIS )
${\displaystyle 24}$	${\displaystyle (24^{n}-1)^{2}-2}$	24, 321, 971, 984, ...
${\displaystyle 26}$	${\displaystyle (26^{n}-1)^{2}-2}$	1, 2, 8, 78, 79, 111, 5276, 8226, 19545, 75993, ...
${\displaystyle 28}$	${\displaystyle (28^{n}-1)^{2}-2}$	1, 2, 11, 15, 586, 993, 5048, 24990, ...
${\displaystyle 30}$	${\displaystyle (30^{n}-1)^{2}-2}$	2, 3, 57, 129, 171, 9837, 30359, 157950, ...
${\displaystyle 32}$	${\displaystyle (32^{n}-1)^{2}-2}$	1, 3, 13, 36, 111, 136, 160, 214, 330, 1273, 7407, 20487, 21276, 22123, 75210, ...
${\displaystyle 34}$	${\displaystyle (34^{n}-1)^{2}-2}$	1, 2, 14, 29, 61, 146, 2901, 6501, 8093, ...
${\displaystyle 36}$	${\displaystyle (36^{n}-1)^{2}-2}$	1, 2, 6, 15, 28, 59, 188, 216, 655, 3884, 4215, 10971, 13465, 16784, 25400, ...
${\displaystyle 38}$	${\displaystyle (38^{n}-1)^{2}-2}$	6, 279, 3490, ...
${\displaystyle 40}$	${\displaystyle (40^{n}-1)^{2}-2}$	2, 49, 144, 825, 2856, 2996, 5166, 7824, 9392, 40778, ...
${\displaystyle 42}$	${\displaystyle (42^{n}-1)^{2}-2}$	1, 3, 4, 81, 119, 2046, 2466, 4020, 7907, 8424, 25002, ...
${\displaystyle 44}$	${\displaystyle (44^{n}-1)^{2}-2}$	3, 195, 1482, 8210, 20502, 60212, 95940, ...
${\displaystyle 46}$	${\displaystyle (46^{n}-1)^{2}-2}$	1, 54, 2040, 3063, ...
${\displaystyle 48}$	${\displaystyle (48^{n}-1)^{2}-2}$	1, 207, 329, 1153, 4687, 13274, 25978, ...
${\displaystyle 50}$	${\displaystyle (50^{n}-1)^{2}-2}$	4, 38, 93, 120, 4396, 11459, 25887, ...

The largest known generalized Kynea prime is and has digits. It was found by Serge Batalov on May 22, 2016 using the programs CKSieve and PrimeFormGW . It is the eighth Kynea prime with this base. ${\displaystyle (30^{157950}+1)^{2}-2}$ ${\displaystyle 466623}$

Further generalizations

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

The smallest prime big ears numbers, so-called big ears prime numbers , are the following:

3, 7, 11, 15, 35, 16475, 26827, 79127, 85075, ... (sequence A0100900 in OEIS )

See also

• Carol 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. ^ Cletus Emmanuel: Message to Yahoo primenumbers group
3. ↑ (2 ⁶⁶¹⁴⁷⁸ +1) ² -2 on The Lagest Known Primes!
4. ↑ ^{a b} Mark Rodenkirch, Gary Barnes, Karsten Bonath: Carol and Kynea Prime Search
5. ↑ (30 ¹⁵⁷⁹⁵⁰ +1) ² -2 on The Lagest Known Primes!
6. ↑ Carol- and Kynea primes