Kynea number

A Kynea number is an integer of the form

4^n + 2^{n + 1} - 1

.

An equivalent formula is

(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:

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

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, 78858, ...
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.

References

↑ Cletus Emmanuel's statement on Yahoo group PrimeNumbers
↑ Entry for 661478th Kynea number at Prime Pages
↑ Carol and Kynea Prime Search by Mark Rodenkirch

External links

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Cletus Emmanuel's statement on Yahoo group PrimeNumbers

[2] Entry for 661478th Kynea number at Prime Pages

[3] Carol and Kynea Prime Search by Mark Rodenkirch

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers Mathematics portal

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