Fortune's conjecture (Fortunate and unfortunate primes : Nearest primes from a prime factorial)

by Cyril Banderier, LIPN, Univ. Paris 13
September 1999 [Updated 12/18/2000, with an addendum in Oct. 2006]

On the distance between the prime factorial and the previous/next prime number

The distribution of prime numbers is one of the most fundamental problems in mathematics. It is nicely related to complex analysis, as e.g. explained in the following lecture note on analytic number theory.

Let us define the "prime factorial of order n" as the product of the first n prime numbers: fn=p1 × p2 × ... × pn.
E.g. f5= 2 × 3 × 5 × 7 × 11= 2310, f6= 2 × 3 × 5 × 7 × 11 × 13=30030. (These numbers fn are also sometimes called "primorial").

When I was a child, I was astonished to read in the Dictionnaire des mathématiques (by Le Lionnais/Bouvier/Georges) that a so-called "conjecture de Fortune" claims that fn+1 is always a prime number. Well, I quickly checked and, in fact f1+1=3, f2+1=7, f3+1=31, f4+1=211, f5+1 =2311, are all primes... but f6+1 = 59 × 509 is certainly not !
Therefore, something was wrong in the way this books presented the conjecture. I will give later the correct conjecture.
Some interesting values of n are those for which fn-1 or fn+1 is prime. These sequences are in fact a way to prove that there exist infinitely many prime numbers as done by Euclid in his nice "reductio ad absurdum".

A natural conjecture (we will explain why hereafter) is the following one

The Prime Factorial Conjecture:

  • The distance dn+ between the prime factorial fn and the first prime greater than fn is either 1, either a prime number.
  • Similarly, the distance dn- between the prime factorial fn and the first prime least than fn is either 1, either a prime number.

    The second line is Banderier's conjecture while the first line is known as Fortune's conjecture (which leads to the so-called "fortunate numbers"). Fortune's conjecture (the first prime greater than a given prime factorial) is studied in the section A2 of the famous book of Richard K. Guy "Unsolved Problems in Number Theory" (2nd edition, Springer, 1994).

    Who was R.F. Fortune ?
    Reo Franklin Fortune (1903-1979) was a social anthropologist, lecturer in social anthropology at the Cambridge University, specialist in Melanesian language and culture (confer the obituaries "Reo FORTUNE (1903-1979)", by Michael W. Young in "Canberra Anthropology" vol. 3, n.1, pp 105-108, 1980). R.F. Fortune joined in 1941 the Department of Anthropology at University of Toronto. Fortune was married to Margaret Mead from 1928 to 1935 (Margaret Mead did have three husbands...the first was Luther Cressman from 1923-28. I believe he was a minister. The second was Reo Fortune, a New Zealand psychologist turned anthropologist from 1928-35. The third was Gregory Bateson (1936-50). He was a British anthropologist whose strong natural science background influenced Mead's work. [Information from a book called Women Anthropologists - Selected Biographies edited by Gacs, Khan, McIntyre, and Weinberg. USA: Greenwood press, 1988]). Fortune is well known for his ethnographies of the Dobu and Manus islanders of the Pacific. What many anthropologists do not realize is that he is also known to mathematicians for his conjecture on prime numbers, or "Fortunate Numbers"! Levin et al. (1984), report a story that Fortune once attempted to conclude an academic dispute with McIlwraith by challenging him to a duel with any weapon of his choice from the collections of the Royal Ontario Museum. Here are other anecdotes kindly communicated to me by Richard L. Warms.

    The reader amazed by this "link" between anthropology and mathematics should keep in mind that is not the only one: remember Claude LÚvi-Strauss, whose mathematical considerations influenced his science and the structuralist school (I consider that this is perhaps in linguistic that structuralism was the most fruitful, with applications in computer sciences to programming languages). Note that the (ab)use of mathematics by Lacan and other people in human sciences was at the origin of the Sokal affair.

    I am not aware of literature about the case of the first prime less than a given prime factorial, so I suggest to call "unfortunate numbers" the dn-'s (whereas the dn+'s are already known as fortunate numbers, as probably coined by M. Gardner, see e.g. The Last Recreations).

    Heuristics: If dn+ were not prime, then dn+=(pn+1)2 or a greater composite number. Asymptocally, one has log( fn )=O(pn/log( pn )) and Shinzel's conjecture (there is always a prime between x and x+ln(x)2) thus implies that the dn's are primes.

    Computations: I give below the list of fortune and unfortunate numbers dn for n<1300,
    previous prime case (EIS A005235),
    next prime case (EIS A055211).

    (EIS refers to the Encyclopedia of integer sequences). Caveat: thus the 2 conjectures are checked for n<1300, provided Algorithm P. (in D.E. Knuth, The Art of Computer Programming, Vol 2, 2nd edition, Section 4.5.4) does not fail when answering that a given number is indeed prime [this is not yet proven for numbers of ten thousands digits]. This is the algorithm used by Maple, it would be interesting to cross-check my results with other computer algebra software (Mathematica, Pari, ...) relying on other primality tests. However, all of them are probabilistic tests (Mathematica uses multiple Rabin-Miller and Lucas, Pari GP uses Baillie-Pomerance-Selfridge-Wagstaff and Lucas) and the best known deterministic test is about 1000 times slower.

    The prime factorial primes, aka as primorial primes (coined by Dubner) are the fn+-1 with
    n such that p1 p2 ... pn+1 is prime : n=1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 67132 [EIS A014545].
    n such that p1 p2 ... pn-1 is prime : n=2, 3, 5, 6, 13, 24, 66 ,68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849 [EIS A057704].

    The factorisation of the composite primorial numbers is given here,

    References:

    [Addendum Oct. 2006:] since the creation of this page, many websites mention the fortunate primes, let me give here some of them : MathWorld, Prime Glossary, primepuzzles.net, a blog...
    T.D. Noe checked the computations in Mathematica for fortunate primes until n=2000, and unfortunate primes until n=1000 and communicated me few typos in my list (see his results on the EIS website).