A Fortunate number, named after Reo Fortune, for a given positive integer n is the smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# 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 pn# is always above pn. This is because pn#, and thus pn# + m, is divisible by the prime factors of m for m = 2 to pn.
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 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 OEIS)).
Reo Fortune conjectured that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number. As of 2012, all the known Fortunate numbers are prime.
Famous quotes containing the words fortunate and/or number:
“It is fortunate to be of high birth, but it is no less so to be of such character that people do not care to know whether you are or are not.”
—Jean De La Bruyère (1645–1696)
“I believe if we introduced the Lord’s Prayer here, senators would propose a large number of amendments to it.”
—Henry Wilson (1812–1875)