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Cullen number - Wikipedia

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Mathematical concept

In mathematics, a Cullen number is a member of the integer sequence C n = n ⋅ 2 n + 1 {\displaystyle C_{n}=n\cdot 2^{n}+1} (where n {\displaystyle n} is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.

Properties

In 1976 Christopher Hooley showed that the natural density of positive integers n ≤ x {\displaystyle n\leq x} for which C_n is a prime is of the order o(x) for x → ∞ {\displaystyle x\to \infty } . In that sense, almost all Cullen numbers are composite.^[1] Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n·2^{n + a} + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for n equal to:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 (sequence A005849 in the OEIS).

Still, it is conjectured that there are infinitely many Cullen primes.

A Cullen number C_n is divisible by p = 2n − 1 if p is a prime number of the form 8k − 3; furthermore, it follows from Fermat's little theorem that if p is an odd prime, then p divides C_m(k) for each m(k) = (2^k − k) (p − 1) − k (for k > 0). It has also been shown that the prime number p divides C_{(p + 1)/2} when the Jacobi symbol (2 | p) is −1, and that p divides C_{(3p − 1)/2} when the Jacobi symbol (2 | p) is + 1.

It is unknown whether there exists a prime number p such that C_p is also prime.

C_p follows the recurrence relation

C p = 4 ( C p − 1 + C p − 2 ) + 1 {\displaystyle C_{p}=4(C_{p-1}+C_{p-2})+1} .

Generalizations

Sometimes, a generalized Cullen number base b is defined to be a number of the form n·bⁿ + 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Cullen prime. Woodall numbers are sometimes called Cullen numbers of the second kind.^[2]

As of April 2025, the largest known generalized Cullen prime is 4052186·69^4052186 + 1. It has 7,451,366 digits and was discovered by a PrimeGrid participant.^[3]^[4]

According to Fermat's little theorem, if there is a prime p such that n is divisible by p − 1 and n + 1 is divisible by p (especially, when n = p − 1) and p does not divide b, then bⁿ must be congruent to 1 mod p (since bⁿ is a power of b^{p − 1} and b^{p − 1} is congruent to 1 mod p). Thus, n·bⁿ + 1 is divisible by p, so it is not prime. For example, if some n congruent to 2 mod 6 (i.e. 2, 8, 14, 20, 26, 32, ...), n·bⁿ + 1 is prime, then b must be divisible by 3 (except b = 1).

The least n such that n·bⁿ + 1 is prime (with question marks if this term is currently unknown) are^[5]^[6]

1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, 2805222, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, 1806676, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, 1341174, 3, ?, 1072, 234, 1, 220, 1, 142, 1295, 8, 3, 16990, 1, 474, 129897, 4052186, 1, 13948, 1, 2525532, 3, 2, 1161, 12198, 1, 682156, 5, 350, 1, 1242, 26, 186, 3, 2, 1, 298, 14, 101670, 9, 2, 775, 202, 1, 1374, 63, 2, 1, ... (sequence A240234 in the OEIS)

b Numbers n such that n × bⁿ + 1 is prime^[5] OEIS sequence 3 2, 8, 32, 54, 114, 414, 1400, 1850, 2848, 4874, 7268, 19290, 337590, 1183414, ... A006552 4 1, 3, 7, 33, 67, 223, 663, 912, 1383, 3777, 3972, 10669, 48375, 1740349, ... A007646 5 1242, 18390, ... 6 1, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496, 515516, ..., 4582770 A242176 7 34, 1980, 9898, 474280, ... A242177 8 5, 17, 23, 1911, 20855, 35945, 42816, ..., 749130, ... A242178 9 2, 12382, 27608, 31330, 117852, ... A265013 10 1, 3, 9, 21, 363, 2161, 4839, 49521, 105994, 207777, ... A007647 11 10, ... 12 1, 8, 247, 3610, 4775, 19789, 187895, 345951, ... A242196 13 ... 14 3, 5, 6, 9, 33, 45, 243, 252, 1798, 2429, 5686, 12509, 42545, 1198433, 1486287, 1909683, ... A242197 15 8, 14, 44, 154, 274, 694, 17426, 59430, ... A242198 16 1, 3, 55, 81, 223, 1227, 3012, 3301, ... A242199 17 19650, 236418, ... 18 1, 3, 21, 23, 842, 1683, 3401, 16839, 49963, 60239, 150940, 155928, 612497, ... A007648 19 6460, ... 20 3, 6207, 8076, 22356, 151456, 793181, 993149, ... A338412

References