Cullen Primes

Last Updated : 23 Jul, 2025

Cullen primes are named after James Cullen, an Irish mathematician who first introduced the concept in 1905. A Cullen number is a number that can be written in the form:

Where, n is any positive integer (1, 2, 3, 4, ...). For first 3 values of n, Cullen numbers are:

Cullen Primes

As we know, a prime number is a number greater than 1 that can only be divided by 1 and itself. For example, 2, 3, 5, 7, 11, . . . etc.

If a Cullen number is prime, it is called a Cullen prime.

Example of Cullen Primes

Let's look at a few Cullen numbers and see which are prime:

For n = 1, C₁ = 3, which is prime. So, 3 is a Cullen prime.
For n = 143, C₁₄₃ = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833, and this is Prime. Thus, for C₁₄₁ is Cullen pime.

The only known Cullen primes are for values of n:

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, and 6679881.

Note: Cullen primes becomes harder as n gets larger because the numbers grow extremely quickly.

Generalized Cullen Prime

A generalized Cullen number base b is defined to be a number of the form n · bⁿ + 1, where n + 2 > b, where n is any natural number. ; if a prime can be written in this form, it is then called a generalized Cullen prime.