IS THE NUMBER OF PRIMES \frac{1}{2} ∑_{i=0}^{n−1} i! FINITE ?

YVES GALLOT

For n ≥ 1 let

!n = ∑_{i=0}^{n−1} i! .

!2 ≡ 0 (mod 2) then !n is divisible by 2 for all n ≥ 2. The numbers !n/2 are prime for n = 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354 and no other n < 500. Are there infinitely many prime numbers of the form !n/2 ? If it is true then p ∤ !p for all prime numbers p > 2. Let

r_{p} = !p mod p

Miodrag Zivković extended the computation of the r_{p} and verified that r_{p} ≠ 0 for 2 < p < 2^{23} [1].

A program was written by the author to extend this computation. The inner loop of the program consists in the simple algorithm

\begin{aligned}f & = (f × i) mod p \\ s & = (s + f) mod p \\ i & = i + 1 \end{aligned} (1)

The floating-point unit is used to replace the slow integer division by a multiplication by 1/p. Because the internal precision of the FP-registers is 64 bits, a value is rounded to the nearest integral value by adding and subtracting the constant 2^{63} + 2^{62}. The value of s is accumulated without modular reduction and only one modular reduction is evaluated outside of the loop. Finally, two different p are checked simultaneously in the same loop to improve the pipelining of the floatingpoint instructions. With these optimizations, the computation time of Alg.1 is 14 cycles on a PII/Celeron/PIII processor.

r_{p} was computed by the program for 2 ≤ p < 2^{26} = 67108864: the results of [1, Table 1] were verified and one new r_{p} such that |r_{p}| < 10 was found: r_{11477429} = 9. No solution to r_{p} = 0 was discovered, then the question of whether the number of primes \frac{1}{2} ∑_{i=0}^{n−1} i! is finite remains an open question.

References

1. M. Zivković, The number of primes ∑_{i=1}^{n}(−1)^{n−i}i! is finite, Math. Comp. 68 (1999), 403–409.

   E-mail address: galloty@wanadoo.fr

Date: July 20, 2000.

2000 Mathematics Subject Classification. Primary 11A41.

Key words and phrases. prime numbers, left factorial.

(C) Copyright 2000, Yves Gallot. You may make unlimited copies of the document and give copies to other persons as long as the copies you make and distribute contain the unaltered and unabridged document.