Carol number

A Carol number is an integer of the form $4^{n} - 2^{n + 1} - 1,$ or equivalently, $(2^{n} - 1)^{2} - 2.$ The first few Carol numbers are: −1, 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527 (sequence A093112 in the OEIS).

The numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.^[1]^[2]

Binary representation

For n > 2, the binary representation of the n-th Carol number is n − 2 consecutive ones, a single zero in the middle, and n + 1 more consecutive ones, or to put it algebraically,

\sum_{i \neq n + 2}^{2 n} 2^{i - 1} .

For example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the 2n-th Mersenne number and the n-th Carol number is $2^{n + 1}$ . This gives yet another equivalent expression for Carol numbers, $(2^{2 n} - 1) - 2^{n + 1}$ . The difference between the n-th Kynea number and the n-th Carol number is the (n + 2)th power of two.

Primes and modular relations

Unsolved problem in mathematics:

Are there infinitely many Carol primes?

(more unsolved problems in mathematics)

Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a prime number, its index n cannot be of the form 3x + 2 for x > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127 (these are listed in Sloane's OEIS: A091516).

The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol emirp.^[3] The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.^[4]

As of April 2020^[update] the largest known prime Carol number has index n = 695631, which has 418812 digits.^[5]^[6] It was found by Mark Rodenkirch in July 2016 using the programs CKSieve and PrimeFormGW. ^[7] It is the 44th Carol prime.

References

↑ Cletus Emmanuel at Prime Pages
↑ Message to Yahoo primenumbers group from Cletus Emmanuel
↑ Prime Curios 16127 at Prime Pages
↑ Prime Curios 16769023 at Prime Pages
↑ Prime Database entry for Carol(695631) at Prime Pages
↑ Carol and Kynea Prime Search Archived 2021-01-26 at the Wayback Machine by Mark Rodenkirch
↑ Carol / Kynea Primes

External links

Weisstein, Eric W. "Near-Square Prime". MathWorld.
Prime Database entry for Carol(695631)
Carol and Kynea Primes Archived 2020-08-25 at the Wayback Machine
Carol and Kynea Prime Search Archived 2021-01-26 at the Wayback Machine

This article "Carol number" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Carol number. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

[1] Cletus Emmanuel at Prime Pages

[2] Message to Yahoo primenumbers group from Cletus Emmanuel

[3] Prime Curios 16127 at Prime Pages

[4] Prime Curios 16769023 at Prime Pages

[5] Prime Database entry for Carol(695631) at Prime Pages

[6] Carol and Kynea Prime Search Archived 2021-01-26 at the Wayback Machine by Mark Rodenkirch

[7] Carol / Kynea Primes

[1]

[2]

[3]

[4]

[5]

[6]

[7]

v t e 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 Tetradic
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

Carol number

Contents

Binary representation

Primes and modular relations

References

External links

📰 Article(s) of the same category(ies)