# Difference between revisions of "Carol-Kynea prime"

## Definitions

In the context of the Carol/Kynea prime search, a Carol number is a number of the form $(b^n-1)^2-2$ and a Kynea number is a number of the form $(b^n+1)^2-2$. A Carol/Kynea prime is a prime which has one of the above forms. A prime of these forms must satisfy the following criteria:

• b must be even, since if it is odd then $(b^n±1)^2-2$ is always even, and thus can’t be prime.
• n must be greater than or equal to 1. For any b, if n is 0 then (bn±1)2 is equal to 1, and thus yields -1 when 2 is subtracted from it. By definition -1 is not prime. If n is negative then (bn±1)2 is not necessarily an integer.
• b may be a perfect power of another integer. However these form a subset of another base’s primes (ex. Base 4 Carol/Kynea primes are Base 2 Carol/Kynea primes where $n \bmod 2 \equiv 0$). So it is not necessary to search these bases separately.

Due to the form of these numbers, they are also classified as near-square numbers (numbers of the form n2-k).

## Top 5 Carol primes

Prime Digits Found by Date
(290124116-1)2-2 611246 Karsten Bonath 2019-03-01
(2695631-1)2-2 418812 Mark Rodenkirch 2016-07-16
(2688042-1)2-2 414243 Mark Rodenkirch 2016-07-05
(17887525-1)2-2 393937 Serge Batalov 2016-05-21
(2653490-1)2-2 393441 Mark Rodenkirch 2016-06-03

## Top 5 Kynea primes

Prime Digits Found by Date
(362133647+1)2-2 683928 Karsten Bonath 2019-06-17
(30157950+1)2-2 466623 Serge Batalov 2016-05-22
(2661478+1)2-2 398250 Mark Rodenkirch 2016-06-18
(196858533+1)2-2 385619 Clint Stillman 2017-11-30
(2621443+1)2-2 374146 Mark Rodenkirch 2016-05-30

## OEIS sequences

These are available OEIS sequences:

Base Carol Kynea
2 A091515 A091513
6 A100901 A100902
10 A100903 A100904
14 A100905 A100906
22 A100907 A100908

## Data

### All bases

All bases with their own page are listed here: There are 381 sequences.

### Bases which are a power of

There are 22 sequences.

### Bases without a Carol prime

There are 84 sequences.

### Bases without a Kynea prime

There are 75 sequences.

### Bases without a Carol and Kynea prime

There are 2 sequences.

### Remaining data

All data not yet given by an own page can be found here.