PAPER 2
The plnp function & the prime number theorem
How to find the approximate value of the nth prime
1.
The prime number theorem
Let n be the number of primes up to a certain size N, then
N
n≈
(1.1)
ln N
and in the limit as N → ∞ we have
N
(1.2)
n = lim
N →∞ ln N
This is the prime number theorem.
2.
The information value of a prime number
Let π be a prime number, then the probability, p, of the prime occurring in an infinite
number system is
1
(2.1)
p=
π
th
For the n prime, the information value, hn , of an event having probability pn is
hn = − pn ln pn
(2.2)
With the n prime π n being the largest number in the system ( π n = N ) we have, from
the Prime Number Theorem
N
π
n=
= n
(2.3)
ln N ln π n
and by rearranging we obtain
n−1 = π n−1 ln π n
th
n −1 = −π n−1 ln π n−1
n −1 = − pn ln pn
(2.4)
By comparing (2.2) with (2.4) we see that
hn = n −1
(2.5)
that is, the information value of the nth prime is the inverse of the number of primes
up to that point.
3.
Determining the size of the nth prime
Using the results from Paper 1, we can determine the approximate size of the nth
prime. As our pn are very small numbers, we are dealing with p1 -type solutions to
(2.2) and
1
pn = −1
(3.1)
−1
hn ln(hn ln(hn−1...))
and so
π n = hn−1 ln(hn−1 ln(hn−1...))
The plnp function & the prime number theorem
and from (2.5)
(3.2)
π n = n ln(n ln(n ln(n...)))
Furthermore, if we call the complementary p2-type solution pn′ then
h
pn pn′ = n
(3.3)
Fn
For large n, however, pn′ → 1 and so
h
π
Fn = n = n
pn n
4.
Conclusions
The following conclusions are approximately true for large n:
The information value of the nth prime, hn, is the inverse of the number of
1.
primes up to and including that prime
2.
The size of the nth prime can be found using
π n = n ln(n ln(n ln(n...)))
3.
The F-numbers will also yield reasonable values where
h
π
Fn = n = n
pn n
and
− Fe ...
p∗ = e− Fe
and where p* is alternately 1 and pn
5.
n
Table of calculated* results and real prime data
pn
π n (calc) π n (real) % Fn
error
10
102
103
2500
104
105
106
107
108
35.8
647.3
9118.0
25351.5
116671.1
1416360
16626509
1.9066x108
2.14881x109
29
541
7919
22307
23
20
15
14
Not
Available
To
Author
3.5771521
6.4727751
9.1180065
10. 140592
11.667115
14.163601
16.626509
19.066002
21.488184
0.0504806
0.0016553
1.10686x10-4
3.96063x10-5
8.58111x10-6
7.06135x10-7
6.01459x10-8
5.24494x10-9
4.65372x10-10
pn′
0.8347884
0.989343
0.9989912
0.9995984
0.9998998
0.99999
0.999999
0.9999999
0.9999999
π n = pn−1
19.8
604.1
9034.5
25248.5
116534.95
1416159.5
16626232
1.90659x108
2.14881x109
* Using an 8-digit calculator
As the percentage error between real and calculated prime values seems to be falling,
it makes one wonder if there is some magically large value for n beyond which the
prime numbers could be predicted with perfect accuracy.
Rob Gough
P a g e |2
March 1985