A fortune number (after the name of New Zealand social anthropologist ) is the smallest integer m > 1 such that for a given positive integer n the number p n # + m is prime , where the primorial p n # is the product of the first n prime numbers.
For example, to find the seventh fortune, you need to calculate the product of the first seven primes (2, 3, 5, 7, 11, 13, and 17), which will give 510510. Adding to result 2 gives again an even number, adding 3 will give a number divisible by 3 , and so it will continue until 18. Addendum 19, however, gives the number 510529, which is simple. So 19 is a fortune number. A fortune number for p n # is always greater than p n and all its divisors are greater than p n . This is a consequence of the fact that p n #, and then p n # + m , are divided by prime divisors of numbers m not exceeding p n .
Fortune numbers for the first few primaries:
- 3 , 5 , 7 , 13 , 23 , 17 , 19 , 23, 37 , 61 , 67 , 61, 71 , 47 , 107 , 59 , 61, 109 , ... (sequence A005235 in OEIS ).
Sorted fortune numbers without repetition:
- 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 ).
Rio Fortun suggested that among these numbers there are no components ( Fortune hypothesis ) [1] . Fortune prime is the number of Fortune, which is also prime; for 2012, all known fortune numbers are prime.
Notes
- ↑ Guy, 1994 , p. 7-8.
Literature
- Chris Caldwell. The Prime Glossary: Fortunate number // Prime Pages .
- Weisstein, Eric W. Fortunate Prime on Wolfram MathWorld .
Richard K. Guy. Unsolved problems in number theory. - 2nd. - Springer, 1994. - S. 7-8. - ISBN 0-387-94289-0 .