Cullen numbers

This article page is a stub, please help by expanding it.

Cullen numbers $C_{n}$ are numbers of the form $n2^{n}+1$. The first few are 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609 (see A002064). Reverend James Cullen noticed that they are almost all composite. The following table gives the prime factorizations of the first few composite Cullen numbers:

$C_{n}$	Prime factorization of $C_{n}$
9	3 2
25	5 2
65	5 × 13
161	7 × 23
385	5 × 7 × 11
897	3 × 13 × 23
2049	3 × 683
4609	11 × 419

Since the Cullen primes (A050920) get very large very quickly, it is easier to write them as $C_{n}$ or $n2^{n}+1$ (see A005849 for the relevant $n$). And when $n$ is even, some people prefer to write them as ${\frac {n}{2^{\alpha }}}2^{n+\alpha }+1$, where $\alpha >0$ is the exponent of 2 in the factorization $n=2^{\alpha }p^{\beta }q^{\gamma }\ldots$. For example, 18496 × 2 18496 + 1 = 17 2 × 2 18502 + 1.

Although Cullen was correct in asserting that almost all Cullen numbers are composite, it remains to be proven whether or not there are infinitely many Cullen primes. It's also unknown if it's possible for both $n$ and $C_{n}$ to be prime.

There are also puzzles concerning the composite Cullen numbers. If $C_{n}$ is prime, then it is trivially true that $\phi (C_{n})\mid C_{n}-1$ (this is called the Lehmer property). A 2011 paper asserts that no composite Cullen number with the Lehmer property exists.

References

• Chris K. Caldwell Cullen primes, The Prime Pages.
• José María Grau Ribas and Florian Luca, "Cullen Numbers with the Lehmer Property", arXiv:1103.3578 [math.NT], Mar 18 2011.