Cullen numbers: C(n) = n·2^n + 1. C(1)=3, C(2)=9, C(3)=25, C(4)=65, C(5)=161... Almost all are composite! Cullen primes are extremely rare: n=1,141,4713,5795,6611,18496... Named after James Cullen (1905).
How to Use
Enter n:
- C(n): Cullen number value
- Prime?: Primality test
- Factors: Small factors
Why So Rare?
Cullen numbers have a covering set of small divisors: C(n) is divisible by 3 when n≡1(mod 2), by 5 when n≡1(mod 4), etc. Multiple divisibility conditions 'cover' most n, leaving very few candidates for primality.
Generalized Cullen
Generalized Cullen: n·b^n+1 for base b>2. These are even rarer as primes. The theory of covering sets explains why: larger bases create more divisibility conditions, leaving fewer prime candidates.
Step-by-Step Instructions
- 1Enter n.
- 2Compute C(n).
- 3Test primality.
- 4Find small factors.
- 5Compare to Woodall.