# Kynea number

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

- .

An equivalent formula is

- .

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.^{[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[edit]

The 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:

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[edit]

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 (sequence A091514 in the OEIS).

As of February 2018^{[update]}, 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[edit]

A **generalized Kynea number base b** is defined to be a number of the form (

*b*

^{n}+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

*b*

^{n}is also a generalized Kynea number to base

*b*.

Least *n* ≥ 1 such that ((2*b*)^{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 (b^{n}+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^{[update]}, the largest known generalized Kynea prime is (30^{157950}+1)^{2} − 2.