# Kynea number

A Kynea number is an integer of the form

${\displaystyle 4^{n}+2^{n+1}-1}$.

An equivalent formula is

${\displaystyle (2^{n}+1)^{2}-2}$.

This indicates that a Kynea number is the nth power of 4 plus the (n + 1)th Mersenne number. Kynea numbers were studied by Cletus Emmanuel who named them after a baby girl.[1]

The sequence of Kynea numbers starts with:

7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, ... (sequence A093069 in the OEIS).

## Properties

The binary representation of the nth Kynea number is a single leading one, followed by n - 1 consecutive zeroes, followed by n + 1 consecutive ones, or to put it algebraically:

${\displaystyle 4^{n}+\sum _{i=0}^{n}2^{i}.}$

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

## Prime Kynea numbers

 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 3x + 1 for x > 0. The first few Kynea numbers that are also prime are 7, 23, 79, 1087, 66047, 263167, 16785407 (sequence A091514 in the OEIS).

As of February 2018, the largest known prime Kynea number has index n = 661478, which has 398250 digits.[2][3] It was found by Mark Rodenkirch in June 2016 using the programs CKSieve and PrimeFormGW. It is the 50th Kynea prime.

## Generalizations

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

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

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, ...
 b numbers n ≥ 1 such that (bn+1)2 − 2 is prime (these n are checked up to 30000) OEIS sequence 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, ... A091513 4 1, 4, 6, 9, 16, 90, 121, 340, 400, 535, 825, 5044, 34656, 53190, 54444, 188025, 221026, 330739, ... 6 1, 2, 3, 4, 9, 12, 30, 49, 56, 115, 118, 376, 432, 1045, 1310, 6529, 7768, 8430, 21942, 26930, 33568, 50800, ... A100902 8 1, 3, 4, 5, 6, 7, 9, 17, 29, 60, 167, 169, 185, 197, 550, 12345, 15291, 23104, 34145, 35460, 36296, 125350, ... 10 22, 351, 1061, ... A100904 12 1, 2, 8, 60, 513, 1047, 7021, 7506, ... 14 1, 5, 60, 72, 118, 181, 245, 310, 498, 820, 962, 2212, 3928, 5844, 5937, ... A100906 16 2, 3, 8, 45, 170, 200, 2522, 17328, 26595, 27222, 110513, ... 18 1, 10, 21, 25, 31, 1083, 40485, ... 20 1, 15, 44, 77, 141, 208, 304, 1169, 3359, 5050, 22431, 34935, ... 22 3, 166, 814, 1851, 2197, 3172, 3865, 19791, ... A100908 24 24, 321, 971, 984, ... 26 1, 2, 8, 78, 79, 111, 5276, 8226, 19545, 75993, ... 28 1, 2, 11, 15, 586, 993, 5048, 24990, ... 30 2, 3, 57, 129, 171, 9837, 30359, 157950, ... 32 1, 3, 13, 36, 111, 136, 160, 214, 330, 1273, 7407, 20487, 21276, 22123, 75210, ... 34 1, 2, 14, 29, 61, 146, 2901, 6501, 8093, ... 36 1, 2, 6, 15, 28, 59, 188, 216, 655, 3884, 4215, 10971, 13465, 16784, 25400, ... 38 6, 279, 3490, ... 40 2, 49, 144, 825, 2856, 2996, 5166, 7824, 9392, 40778, ... 42 1, 3, 4, 81, 119, 2046, 2466, 4020, 7907, 8424, 25002, ... 44 3, 195, 1482, 8210, 20502, 60212, 95940, ... 46 1, 54, 2040, 3063, ... 48 1, 207, 329, 1153, 4687, 13274, 25978, ... 50 4, 38, 93, 120, 4396, 11459, 25887, ...

As of February 2018, the largest known generalized Kynea prime is (30157950+1)2 − 2.