Cullen number
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In mathematics, a Cullen number is a member of the
integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
C_n = n \cdot 2^n + 1 (where n is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
). 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 ...
s.


Properties

In 1976
Christopher Hooley Christopher Hooley (7 August 1928 – 13 December 2018) was a British mathematician, professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in ...
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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s n \leq x for which ''C''''n'' is a
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
is of the order ''o''(''x'') for x \to \infty. In that sense, almost all Cullen numbers are
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
. 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 number In number theory, a Woodall number (''W'n'') is any natural number of the form :W_n = n \cdot 2^n - 1 for some natural number ''n''. The first few Woodall numbers are: :1, 7, 23, 63, 159, 383, 895, … . History Woodall numbers were first st ...
s. 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 conjectured that there are infinitely many Cullen primes. A Cullen number ''C''''n'' is
divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
by ''p'' = 2''n'' − 1 if ''p'' is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of the form 8''k'' − 3; furthermore, it follows from Fermat's little theorem 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 Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
(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. ''Cp'' follows the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
: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 number In number theory, a Woodall number (''W'n'') is any natural number of the form :W_n = n \cdot 2^n - 1 for some natural number ''n''. The first few Woodall numbers are: :1, 7, 23, 63, 159, 383, 895, … . History Woodall numbers were first st ...
s are sometimes called Cullen numbers of the second kind. As of October 2021, the largest known generalized Cullen prime is 2525532·732525532 + 1. It has 4,705,888 digits and was discovered by Tom Greer, a
PrimeGrid PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...
participant. 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 ''b''''n'' must be
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
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, ...


References


Further reading

* . * . * . * .


External links

* Chris Caldwell
The Top Twenty: Cullen primes
at The
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
.
The Prime Glossary: Cullen number
at The Prime Pages. * Chris Caldwell
The Top Twenty: Generalized Cullen
at The Prime Pages. *

(outdated), Cullen Prime Search is now hosted at
PrimeGrid PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...
* Paul Leyland
(Generalized) Cullen and Woodall Numbers
{{Classes of natural numbers __NOTOC__ Integer sequences Unsolved problems in number theory Classes of prime numbers