Fortunate number

Unsolved problem in mathematics:

Are any Fortunate numbers composite? (Fortune's conjecture)

In number theory, 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. If a composite Fortunate number does exist, it must be greater than or equal to p_n+1². ^{[citation needed]}

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, etc. (sequence A005235 in the OEIS).

The Fortunate numbers sorted in numerical order with duplicates removed:

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, 199, ... (sequence A046066 in the OEIS).

Fortune conjectured that no Fortunate number is composite (Fortune's conjecture).^[1] A Fortunate prime is a Fortunate number which is also a prime number. As of 2017^[update], all known Fortunate numbers are prime, checked up to n=3000.

References

^ Guy, Richard K. (1994). Unsolved problems in number theory (2nd ed.). Springer. pp. 7–8. ISBN 0-387-94289-0.

Chris Caldwell, "The Prime Glossary: Fortunate number" at the Prime Pages.
Weisstein, Eric W. "Fortunate Prime". MathWorld.

[1] Guy, Richard K. (1994). Unsolved problems in number theory (2nd ed.). Springer. pp. 7–8. ISBN 0-387-94289-0.

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