Fortunate number

A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, p_n# + m is a prime number, where the primorial p_n# is the product of the first n prime numbers.

For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18. Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for p_n# is always above p_n and all its divisors are larger than p_n. This is because p_n#, and thus p_n# + m, is divisible by the prime factors of m not larger than p_n.

The Fortunate numbers for the first primorials are:

The Fortunate numbers sorted in numerical order with duplicates removed:

Reo Fortune conjectured that no Fortunate number is composite (Fortune's conjecture). A Fortunate prime is a Fortunate number which is also a prime number. As of 2012^[update], all the known Fortunate numbers are prime.

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