Fortunate number

Template:Short description Template:Distinguish Template:Unsolved In number theory, a Fortunate number 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. They are named after Reo Fortune.

For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes, which is

2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 = 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 numbers for the first primorials are:

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, \dots

Template:OEIS. The Fortunate numbers sorted in numerical order with duplicates removed are

3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, \dots

Template:OEIS.

Fortune conjectured that no Fortunate number is composite.Template:Sfnp A Fortunate prime is a Fortunate number which is also a prime number. Template:As of, all known Fortunate numbers are prime, checked up to $n = 3000$ .

The Fortunate number for $p_{n} #$ is always above $p_{n}$ and all its divisors are larger than $p_{n}$ . This is because $p_{n} # + m$ is divisible by the prime factors of $m$ not larger than $p_{n}$ . It follows that if a composite Fortunate number does exist, it must be greater than or equal to $p_{n + 1}^{2}$ .^[1]

Paul Carpenter defines the less Fortunate numbers as the differences between $p_{n} #$ and the largest prime less than $p_{n} # - 1$ . These are also conjectured to be always prime.^[1]