Problems & Puzzles:
R. F Fortune conjetured that Nk = P - pk# =
primes, if P = the least prime bigger than (pk# +1), being pk#=126.96.36.199.
As far as I understood, this is a curious procedure to get a safe
small prime from several other smaller primes (pk#)and a bigger one (P).
If you try to calculate the "Fortunate" primes youll
observe that :
- you do not produce all the natural prime numbers (i.e. 2, 11, 29, 31,
41, 43, 53, 73, etc.)
- sometimes you produce twice the same prime number (i.e. N5
& N8 are 23)
- while at first you can easily calculate the Fortunate primes, soon
its 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
(Ref. 2, P. 7)
Said el aidi wrote (10/08/2000):
tackle the Fortune conjecture, it’s necessary and it’s enough to
show the following difficult conjecture: pi(
means that: « for
always a prime between x+1
a little finer than that conjectured by Schinzel: pi(
which link exists between our conjecture and the Fortune one ?. Let
us start by supposing true
our conjecture, we obtain for
1975, Rosser & Schoenfeld
that : ln(pk#)
It’s easy to show that (1.001102)199
pk , for k>2
Nk is composite, then it must have a prime factor lesser than
pk+1 ( 2, 3, 5, 7, …, or pk ). Therefore
is composite, this contradicts the fact that P is prime .Then Nk
must be prime.
k= 1, 2 we can checked the validity of the Fortune’s conjecture
by a simple numerical calculation.
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 .
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.
Teta(x) is called the Chebyshev’s function, and I
used it for x= pk."
Luis Rodríguez sent (October 2003) the
following comment to the Said's contribution:
I read your heuristic argument in favor of
Fortune's Conjecture, but unfortunately I found a difficulty in your
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 =
In this case there is not limit for n -->
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) =
the actual Delta is 1132 and, your
admissible Delta is 1186.