Problems & Puzzles: Problems

Problem 4.- Fortune's Conjecture

R. F Fortune conjetured that N_k = P - p_k# = primes, if P = the least prime bigger than (p_k# +1), being p_k#=2.3.5.7. … .pk

As far as I understood, this is a curious procedure to get a safe small prime from several other smaller primes (p_k#)and a bigger one (P).

If you try to calculate the "Fortunate" primes you’ll observe that :

1. you do not produce all the natural prime numbers (i.e. 2, 11, 29, 31, 41, 43, 53, 73, etc.)
2. sometimes you produce twice the same prime number (i.e. N₅ & N₈ are 23)
3. while at first you can easily calculate the Fortunate primes, soon it’s a harder & harder (time consuming) task.

This are the first 20 Fortunate primes : 3, 5, 7, 13, 23,17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103.

What exactly is the Fortunate procedure ? Is this the first and unique known "mechanical" procedure to produce as many prime number (forget for a moment the difficulties) as you want. Do you have an explanation or a comment for this conjecture?

(Ref. 2, P. 7)

Solution

Said el aidi wrote (10/08/2000):

"To tackle the Fortune conjecture, it’s necessary and it’s enough to show the following difficult conjecture: pi( x+(lnx)^1.99 ) > pi(x+1), for x>8 which means that: « for x>8, there is always a prime between x+1 and  x+(lnx)^1.99  ».

It’s a little finer than that conjectured by Schinzel: pi( x+(lnx)² ) > pi(x)    for x>8

But which link exists between our conjecture and the Fortune one ?. Let us start by supposing  true our conjecture, we obtain for

x =p_k# =2*3*5*…*p_k , k>2 :

p_k# +1< P = prime < p_k# +(ln(p_k#))^1.99

i.e 1< N_k = P - p_k# < (ln(p_k#))^1.99

In 1975, Rosser & Schoenfeld showed that : ln(p_k#) < 1.001102p_k, hence

1 < N_k < (ln(p_k#))^1.99 < (1.001102p_k)^1.99 < p²_k< p²_k+1

( It’s easy to show that  (1.001102)¹⁹⁹ < p_k  , for k>2 )

Suppose N_k is composite, then it must have a prime factor lesser than p_k+1 ( 2, 3, 5, 7, …, or p_k ). Therefore P = N_k + p_k#  is composite, this contradicts the fact that P is prime .Then N_k must be prime. For k= 1, 2 we can checked the validity of the Fortune’s conjecture   by a simple numerical calculation.

I believe that the procedure of Fortune and that of Carl Pomerance  that I generalized (see Problem 2) are the only ones which can existed and which can give primes, but since the time I can’t prove that .

The Rosser & Schoenfeld relation is a straight proof & the formal  reference of this work is Rosser, J.B. & Schoenfeld, L. Sharper bounds for Chebyshev functions Teta(x) and PHI(x). Math .Comp., 29, 243-269,1975.

Note: Teta(x) is called the Chebyshev’s function, and I used it for x= pk."

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Luis Rodríguez sent (October  2003) the following comment to the Said's contribution:

Dear Mr. Said

I read your heuristic argument in favor of Fortune's Conjecture, but unfortunately I found a difficulty in your Cramer's modification.

Calling Delta = p(n+1) - p(n) , Cramer affirms:

Lim Sup. Delta / (Log(p(n))^2 = 1 . But if we multiply both sides by (Log(p(n))^.0.01

Lim Sup. Delta / (Log(p(n))^1.99 = (Log(p(n))^0.01

In this case there is not limit for n --> infinite.

Besides I suspect that sooner than later that modification  will be contradicted by a counterexample. In 1999 Bertil Nyman was very near with p(n) = 1.693....x 10^15 the actual Delta is 1132 and, your admissible Delta is 1186.

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