Cullen number

In 1976 Christopher Hooley showed that the natural density of positive integers $n\; \backslash leq\; x$ for which C_n is a prime is of the order o(x) for $x\; \backslash to\; \backslash infty$. In that sense, almost all Cullen numbers are composite.{{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | author2-link=Alfred van der Poorten | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=Providence, RI | publisher=American Mathematical Society | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 | page=94 }} 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 {{OEIS|id=A005849}}.

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

:$C\_p=4(C\_\{p-1\}+C\_\{p-2\})+1$.

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.{{Cite journal|last=Marques|first=Diego|year=2014|title=On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers|url=https://cs.uwaterloo.ca/journals/JIS/VOL17/Marques/marques5r2.pdf|journal=Journal of Integer Sequences|volume=17}}

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.{{Cite web|url=https://www.primegrid.com/download/GC69-4052186.pdf|title=PrimeGrid Official Announcement|date=26 April 2025|website=Primegrid|access-date=26 April 2025}}{{cite web|title=PrimePage Primes: 4052186 · 69^4052186 + 1|url=https://t5k.org/primes/page.php?id=140607|access-date=26 April 2025|website=t5k.org}}

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{{cite web|url=http://guenter.loeh.name/gc/status.html |title=Generalized Cullen primes |date=6 May 2017 |last=Löh |first=Günter }}{{cite web|url=http://harvey563.tripod.com/GClist.txt |title=List of generalized Cullen primes base 101 to 10000 |date=6 May 2017 |last=Harvey |first=Steven }}

: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, ... {{OEIS|id=A240234}}

{{Reflist}}

• {{Citation |last=Cullen |first=James |title=Question 15897 |journal=Educ. Times |date=December 1905 |page=534}}.
• {{Citation |first=Richard K. |last=Guy |author-link=Richard K. Guy |title=Unsolved Problems in Number Theory |edition=3rd |publisher=Springer Verlag |location=New York |year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | at=Section B20 }}.
• {{Citation |last=Hooley |first=Christopher |author-link=Christopher Hooley |title=Applications of sieve methods |publisher=Cambridge University Press |year=1976 |isbn=0-521-20915-3 |pages=115–119 | zbl=0327.10044 | series=Cambridge Tracts in Mathematics | volume=70 }}.
• {{Citation |first=Wilfrid |last=Keller |title=New Cullen Primes |journal=Mathematics of Computation |volume=64 |issue=212 |year=1995 |pages=1733–1741, S39–S46 |url=https://www.ams.org/mcom/1995-64-212/S0025-5718-1995-1308456-3/S0025-5718-1995-1308456-3.pdf | zbl=0851.11003 | issn=0025-5718 |doi=10.2307/2153382|jstor=2153382 |doi-access=free }}.

• Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=6 The Top Twenty: Cullen primes] at The Prime Pages.
• [http://primes.utm.edu/glossary/page.php?sort=Cullens The Prime Glossary: Cullen number] at The Prime Pages.
• Chris Caldwell, [https://primes.utm.edu/top20/page.php?id=42 The Top Twenty: Generalized Cullen] at The Prime Pages.
• {{MathWorld | urlname=CullenNumber | title=Cullen number}}
• [https://web.archive.org/web/20220318135759/http://www.prothsearch.com/cullen.html Cullen prime: definition and status] (outdated), Cullen Prime Search is now hosted at PrimeGrid
• Paul Leyland, [https://web.archive.org/web/20131219031355/http://www.leyland.vispa.com/numth/factorization/cullen_woodall/cw.html (Generalized) Cullen and Woodall Numbers]

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Category:Integer sequences

Category:Unsolved problems in number theory

Category:Classes of prime numbers