Unsolved problems in mathematics: Are any Fortunate numbers composite?

A Fortunate number, named after Reo Fortune, for a given positive integer n is the smallest integer m > 1 such that p_n# + m is a prime number, where the primorial p_n# is the product of the first n prime numbers. Reo Franklin Fortune (1903&ndash1979 was a New Zealand social anthropologist lecturer in social anthropology at the Cambridge University, specialist in Melanesian language The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The primorial has two similar but distinct meanings The name is attributed to Harvey Dubner and is a Portmanteau of prime and Factorial

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. This is because p_n#, and thus p_n# + m, is divisible by the prime factors of m for m = 2 to p_n. In Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder

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. ---- In mathematics Three is the first odd Prime number, and the second smallest prime This article discusses the number five. For the year 5 AD see 5. In mathematics Seven is the fourth Prime number. It is not only a Mersenne prime (since 23 &minus 1 = 7 but also a This article is about the number 23 For the year see 23. For the movies see 23 (film and The Number 23. 17 ( seventeen) is the Natural number following 16 and preceding 18. 19 ( nineteen) is the Natural number following 18 and preceding 20. 37 ( thirty-seven) is the Natural number following 36 and preceding 38. 61 ( sixty-one) is the Natural number following 60 and preceding 62. 67 ( sixty-seven) is the Natural number following 66 and preceding 68. 71 ( seventy-one) is the Natural number following 70 and preceding 72. 47 ( forty-seven) is the Natural number following 46 and preceding 48. 59 ( fifty-nine) is the Natural number following 58 and preceding 60. (sequence A005235 in OEIS). The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

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 (A046066).

Reo Franklin Fortune (1903–1979), an anthropologist who was married to Margaret Mead, was the first to discover such numbers. Reo Franklin Fortune (1903&ndash1979 was a New Zealand social anthropologist lecturer in social anthropology at the Cambridge University, specialist in Melanesian language Margaret Mead ( December 16, 1901, Philadelphia &ndash November 15, 1978, New York City) was an American He also conjectured that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number. As of 2008, all the known Fortunate numbers are also Fortunate primes. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common

References

• Chris Caldwell, "The Prime Glossary: Fortunate number" at the Prime Pages. The Prime Pages is a website about Prime numbers maintained by Prof
• Eric W. Weisstein, Fortunate Prime at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W