# 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 2006, the largest known prime Kynea number has index *n* = 281621 and approximately equals 5.5×10^{169552}. It was found by Cletus Emmanuel in November 2005, using *k*-Sieve from Phil Carmody and OpenPFGW. This is the 46th 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 *k* ≥ 1 such that ((2*n*)^{k}+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 20000) |
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, 221025, 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, ... | |

18 | 1, 10, 21, 25, 31, 1083, ... | |

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, ... | |

28 | 1, 2, 11, 15, 586, 993, 5048, 24990, ... | |

30 | 2, 3, 57, 129, 171, 9837, 30359, 157950, ... |

As of September 2017^{[update]}, the largest known generalized Kynea prime is (30^{157950}+1)^{2} − 2.