Fortunate number

In number theory, a Fortunate number is the smallest integer ${\displaystyle m>1}$ such that, for a given positive integer ${\displaystyle n}$, ${\displaystyle p_{n}\#+m}$ is a prime number, where the primorial ${\displaystyle p_{n}\#}$ is the product of the first ${\displaystyle n}$ 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

${\displaystyle 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17=510510.}$

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:

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

(sequence A005235 in the OEIS). The Fortunate numbers sorted in numerical order with duplicates removed are

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

(sequence A046066 in the OEIS).

Fortune conjectured that no Fortunate number is composite.^[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 ${\displaystyle n=3000}$.

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

Paul Carpenter defines the less Fortunate numbers as the differences between ${\displaystyle p_{n}\#}$ and the largest prime less than ${\displaystyle p_{n}\#-1}$. These are also conjectured to be always prime.^[2]

• Guy, Richard K. (1994) [1981]. Unsolved Problems in Number Theory. Problem Books in Mathematics. Vol. 1 (2nd ed.). New York: Springer. ISBN 0-387-94289-0.

• Weisstein, Eric W. "Fortunate Prime". MathWorld.

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