Re: Towards a Formula for Primes

On May 11, 2:30 pm, "charlesweh...@xxxxxxxxxxx"
<charlesweh...@xxxxxxxxxxx> wrote:
Before reading on, be warned that this Newsgroup is notorious for its
bickering.

It is unfortunate that it is not moderated. The issues are beyond the
conception of the bickerers, and so they assume that they are beyond
concetion itself.

I am not the only target of this MINDLESS abuse. Douglas Eagleson has
obviously been exercising a fine mind, but when he decides to deliver
his ideas there seems to be a language barrier. Perhaps English is not
his mother tongue.

I followed an AXIOMATIC approach, and applied the GÖDEL INCOMPLETENESS
THEOREM (as a tool - I did not prove it because Gödel did that). This
led me to decide that at least one logic function would be needed
together with arithmetic.

Overnight I have now written a simple "Sieve of Eratosthenes" program
in Qbasic. It segregates the primes from the non-primes. There are, I
can report, twenty-five primes below 100. So it is exactly one in
four.

I had done work on primes together with Alan Coutanceau-Clark years
ago. He was about to deliver a lecture to the Royal Society when he
suddenly died. A fine mind was lost to the world. However, I remember
that the primes up to 64k were about one in ten.

As the primes get higher, they get rarer. I had previously stated that
at the level of a googolplex (ten to the ten to the ten, or a hundred-
figure number), they would be millions of millions apart. This was an
understatement. A malicious bickerer said it would be one percent.
RUBBISH.

The Qbasic program acts rather like a "comb filter". You create a
virtual "comb" of infinite length with its tines ("teeth") two units
apart, and "comb out" all even numbers above two - starting with 4 -
from a "number line" with numbers one unit apart.

In the Qbasic program, which reaches a hundred, we use a "comb" of
pitch three, then pitch four and so on up to ten. We stop at the
square root. The output of that program is given below.

If it had been a googolplex, we would have had to comb ten to the ten
to the five times - a hundred million million million million million
million million million times. There will be almost nothing left at
the level of a googolplex. One percent is a RISIBLE estimate.

So beware those on this Newsgroup who give you "opinions". Believe
only facts.

I created this simple Qbasic program in just two minutes, to get some
primes to study their bit-pattern in the hope of spotting something
unique. The non-primes are segregated out and shown, so that counter-
examples can be tested. Any property of primes must be absent from non-
primes.

Mathematics, not opinion, will reveal the truth.

"Mathematics is the closest you can get to God" - Alan
Coutanceau.Clark, atheist.

Charles Douglas Wehner

------------------------------------------------------------
PROGRAM

DIM a%(100)

FOR n = 2 TO 100
a%(n) = -1
NEXT n

FOR n = 2 TO 10
m = n + n
FOR m = m TO 100 STEP n
a%(m) = 0
NEXT m
NEXT n

PRINT

FOR n = 1 TO 100
IF a%(n) THEN PRINT n,
NEXT n

PRINT

FOR n = 1 TO 100
IF NOT a%(n) THEN PRINT n,
NEXT n

OUTPUT

2 3 5 7 11
13 17 19 23 29
31 37 41 43 47
53 59 61 67 71
73 79 83 89 97

1 4 6 8 9
10 12 14 15 16
18 20 21 22 24
25 26 27 28 30
32 33 34 35 36
38 39 40 42 44
45 46 48 49 50
51 52 54 55 56
57 58 60 62 63
64 65 66 68 69
70 72 74 75 76
77 78 80 81 82
84 85 86 87 88
90 91 92 93 94
95 96 98 99 100

The Prime Number Theorem tells us that the number of primes less than
n is approximately equal to the logrithmic integral:

pi(n) ~= Li(n) = integral from 2 to n {dx / ln(x)}

See http://mathworld.wolfram.com/PrimeNumberTheorem.html

The derivative of Li(n) is the density of primes in the neigborhood of
n:

d(n) = 1/ln(n)

You had stated that for a "hundred-figure number" the primes would be
"millions of millions apart". Gerry Myerson corrected you and said
"for 100-digit numbers the difference between
consecutive primes is, on average, only a few hundred". His statement
is based directly on the Prime Number Theorem. In the post just above
you say you were refering to a googleplex. But for a googleplex =
10^google the distance between primes about about a few google, far
more than millions of millions.

- MO

.