September 1999 [Updated 12/18/2000, with an addendum in Oct. 2006]

Let us define the "prime factorial of order *n*"
as the product of the first *n* prime numbers:
*f _{n}=p_{1} × p_{2} ×
... × p_{n}*.

E.g.

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 f_{n}+1 is always a prime number.
Well, I quickly checked and, in fact
f_{1}+1=3,
f_{2}+1=7,
f_{3}+1=31,
f_{4}+1=211,
f_{5}+1 =2311, are all primes... but f_{6}+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
*f _{n}-*1 or

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

**The Prime Factorial Conjecture:**

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 Lévi-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
*d _{n}^{-}*'s
(whereas the

**Heuristics:**
If *d _{n}^{+}*
were not prime, then

**Computations:**
I give below the list of fortune and unfortunate numbers
*d _{n}* for

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 *f _{n}+-1* with

The factorisation of the composite primorial numbers is given here,

**References:**

- Chris K. Caldwell, Yves Gallot, On The Primality of n!+- 1 and 2.3.5...p+- 1, Math. Comp. 71 (2002), 441-448
- H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203.
- A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, 1962, Vol. 1, p. 50.
- S. W. Golomb, The evidence for Fortune's conjecture, Math. Mag. 54 (1981), 209-210.
- P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
- H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.

[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).