mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Cullen number is a member of the integer sequence

C_n = n \cdot 2^n + 1

(where

n

is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of

Proth number A Proth number is a natural number ''N'' of the form N = k \times 2^n +1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François ...

Properties

In 1976 Christopher Hooley showed that the

natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...

of positive integers

n \leq x

for which ''C''_''n'' is a prime is of the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

''o''(''x'') for

x \to \infty

. In that sense,

almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...

Cullen numbers are composite. 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 . Still, it is

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

d that there are infinitely many Cullen primes. A Cullen number ''C''_''n'' is divisible by ''p'' = 2''n'' − 1 if ''p'' is a prime number of the form 8''k'' − 3; furthermore, it follows from

Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...

that if ''p'' is an

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

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''_{(3''p'' − 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_+C_)+1

Generalizations

Sometimes, a generalized Cullen number base ''b'' is defined to be a number of the form ''n''·''b''^''n'' + 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. As of October 2021, the largest known generalized Cullen prime is 2525532·73^2525532 + 1. It has 4,705,888 digits and was discovered by Tom Greer, a PrimeGrid participant. According to

, 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 ''b''^''n'' must be congruent to 1 mod ''p'' (since ''b''^''n'' is a power of ''b''^{''p'' − 1} and ''b''^{''p'' − 1} is congruent to 1 mod ''p''). Thus, ''n''·''b''^''n'' + 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''·''b''^''n'' + 1 is prime, then ''b'' must be divisible by 3 (except ''b'' = 1). The least ''n'' such that ''n''·''b''^''n'' + 1 is prime (with question marks if this term is currently unknown) are :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, ?, 1, 13948, 1, ?, 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, ...

Properties

Generalizations

References

Further reading

External links