Cullen Primes (original) (raw)
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:
C_n = n \cdot 2^n + 1
Where, n is any positive integer (1, 2, 3, 4, ...). For first 3 values of n, Cullen numbers are:
- C_1 = 1 \cdot 2^1 + 1=2+1=3
- C_2 = 2 \cdot 2^2 + 1 = 2 \cdot 4 + 1 = 8 + 1 = 9
- C_3 = 3 \cdot 2^3 + 1 = 3 \cdot 8 + 1 = 24 + 1 = 25
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 ยท _b n + 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.
Summary
- A **Cullen number is a number in the form C_n = n \cdot 2^n + 1.
- A **Cullen prime is a Cullen number that is also a prime number.
- Cullen primes are rare and grow very large as n increases.
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