Let me summarize very briefly the argument in my previous post.
For any [italics[known[italics], countably [italics]finite[italics] set of
primes with each prime in the set being < x, with x a large positive real
number, Allen's algorithm will work, especially if he uses the Prime Number
Theorem function Pi(x) in his algorithm for some finite number of primes. But I
have good reason to believe there is no such algorithm for the whole countably
infinite set of primes
A = {2, 3, 5, 7, 11, 13, ...}
Assume that last statement is false. Then assume F is Allen's iterative
algorithm that inputs any prime p_{i} in A and outputs the integer sequence
term i in the sequence of infinite primes in A, such that
p_{i} ---------> F(p_{i}) = i.
This obviously is a 1--1 onto map from the countably infinite set of primes to
the countably infinite set of natural numbers. In real analysis one knows that
some 1--1 onto inverse map F^{-1} must exist too for F, if F as an iterative
map on all the elements in A really by assumption does exist. Then it follows
from assumption about the existence of F that, for some 1--1 onto inverse map
F^{-1} we have
F^{-1}: i -------> p_{i}.
This also is 1--1 and onto if F really exists (See "Topology" Second Edition,
2000, by Munkres, Chapter 1, page 18). Then if by assumption the iterative
algorithm function F really exists then it follows from that assumption so does
F^{-1}.
Then there exists some inverse 1--1, onto iterative algorithm F^{-1}:
F^{-1}: i ---------> p_{i},
that cranks out any prime number p_{i} as OUTPUT for any INPUT sequence term
number i. And quite a few illustrious mathematicians living and dead have said
any algorithm like F^{-1} is just impossible, and so the assumption in the
beginning leads to a contradiction. So if the inverse iterative algorithm
F^{-1}
cannot exist neither can F. Algorithms like F and F^{-1} would have to be
deterministic, but the entire countably infinite set of primes A = {2, 3, 5,
...}, has a definite random or stochastic nature among its elements that still
is not completely understood.
It's funny that the Riemann zeta function zeta(z) = 1/1^{z} + 1/2^{z} + 1/3^{z}
+ ... is important in the Riemann Hypothesis and in some problems in
statistical mechanics at the same time.
But that does NOT mean Allen cannot construct some workable algorithm that can
output the number i for a prime p_{i} appearing as input and in some very large
but finite set of primes
{2, 3, 5, ..., p}, with p >> 1,
where each prime in the set is < x for positive very large real number x such
that Pi(x) is known, and each prime in the set is known. So if he is talking
about a finite set of primes his algorithm works, I say yes, he's right and
good luck with publication.
:-(
Robert Betts
Graduate Student
(Alumnus, 2002): University of Massachusetts
Department of Mathematics and Science
Science Building
100 Morrissey Boulevard
Boston, MA USA 02125
[Post refers to previous posts/threads following below]
....................................................................................................................................................................
-----Forwarded Message-----
>From: "[EMAIL PROTECTED]" <[EMAIL PROTECTED]>
>Sent: Oct 2, 2006 2:26 PM
>To: [email protected]
>Cc: [EMAIL PROTECTED]
>Subject: Re: Prime Digest, Vol 30, Issue 1
>
>[Previous]
>
>
>Message: 5
>>Date: Sun, 01 Oct 2006 09:22:10 +0200
>>From: [EMAIL PROTECTED]
>>Subject: [Prime] Possible algorithm for predicting the sequence number
>> for each prime?
>>To: [email protected]
>>Cc: [EMAIL PROTECTED]
>>Message-ID: <[EMAIL PROTECTED]>
>>Content-Type: text/plain; charset=ISO-8859-1
>>
>>Hi,
>>
>>Robert Betts said: "I meant to add that all the prime numbers are known to be
>>distributed randomly--I repeat--randomly along the real line."
>>
>>Why "randomly" ?
>>If I remember well, a set of numbers is random if the shortest way to describe
>>it is to provide the list of these numbers (there is no algorithm to compute
>>them, and knowing the first N numbers does not help to predict number numbered
>>N+1).
>>Since the Eratosthem sieve algorithm can produce the list of all prime
>>numbers,
>>prime numbers do not appear randomly.
>>Since knowing all primes below sqrt(P) can be used to prove that P is prime or
>>not, prime numbers do not appear randomly.
>>Large prime numbers seems to appear randomly to us because they would require
>>computers and time as big as our Universe.
>>
>>So: prime numbers are not distributed randomly.
>>
>>See:
>>http://en.wikipedia.org/wiki/Random_number
>>http://en.wikipedia.org/wiki/Chaitin%E2%80%93Kolmogorov_randomness
>>http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html
>>
>>Regards,
>>
>>Tony
>>
>>
>>------------------------------
>>
>>Message: 6
>>Date: Sun, 1 Oct 2006 17:40:55 +0000
>>From: Brian Beesley <[EMAIL PROTECTED]>
>>Subject: Re: [Prime] Possible algorithm for predicting the sequence
>> number for each prime?
>>To: The Great Internet Mersenne Prime Search list <[email protected]>
>>Message-ID: <[EMAIL PROTECTED]>
>>Content-Type: text/plain; charset="iso-8859-1"
>>
>>On Sunday 01 October 2006 07:22, [EMAIL PROTECTED] wrote:
>>
>>> Robert Betts said: "I meant to add that all the prime numbers are known to
>>> be distributed randomly--I repeat--randomly along the real line."
>>
>>That's obviously wrong - though (over reasonable intervals) the density of
>>prime numbers is approximately uniformly distributed when plotted against the
>>_logarithm_ of the interval midpoint.
>>>
>>> Why "randomly" ?
>>> If I remember well, a set of numbers is random if the shortest way to
>>> describe it is to provide the list of these numbers (there is no algorithm
>>> to compute them, and knowing the first N numbers does not help to predict
>>> number numbered N+1).
>
>
>
>............................................................................................................................................................
>
>[Response]
>
>By the term below "sequence number," I mean 1 is the sequence number for the
>prime 2, and 2 for 3, and 3 for 5, 4 for 7, 5 for 11, 6 for 13, and so on,
>where the sequence is
>
>
>2, 3, 5, 7, 11, 13, 17, ....
>
>
>
>If Poapst's algorithm uses the Prime Number Theorem function Pi(x) for large
>input number prime p, then a subroutine or other algorithm could assign a
>number for p to give its position in the prime number sequence
>
>
>2, 3, 5, ..., p,
>
>
>provided each and every prime less than p is known. Therefore I agree that if
>A. Poapst's algorithm does use the Prime Number Theorem, he could add an
>additional algorithm that could assign a probable sequence number for p, but
>GIVEN that ALL the primes < p also are known. But what if they are NOT known?
>How can Poapst's algorithm be used to assign sequence numbers for primes that
>are not yet known to exist?
>
>
>However, even this--using the Prime Number Theorem--does not show that the
>primes are not randomly distributed along the real line. If prime numbers were
>not randomly distributed along the real line, and if I am not mistaken on
>this, a lot of unsolved conjectures would be answered. Any prime easily could
>be found if they are not randomly distributed, including the twin primes.
>
>
>Let me illustrate. Suppose someone was to say, "What is the 1000-th positive
>even integer? That's easy. Just use an algorithm
>
>
>
>Each Even Number = 2n, n a positive integer,
>
>
>and plug in 1000 as n in the iteration, you get 2000 as the 1000-th positive
>even integer.
>
>
>But can you do this with all prime numbers?
>
>
>Let me raise the issue on this List as a formal question:
>
>
>QUESTION: Does there exist an iteration algorithm F for any input prime p,
>such that, for any such prime number p appearing in the sequence,
>
>
>
>2, 3, 5, 7, 11, 13, 17, ..., p, ...,
>
>we have
>
>p ------------> F(p) = i,
>
>
>where "i" is the position of the prime number p in the sequence 2, 3, 5, ....?
>
>
>What about without using the Prime Number Theorem? If the primes are not
>randomly distributed on the real line, then it would seem you would not need
>the Prime Number Theorem function Pi() to find out what number of prime the
>prime p is in the infinite sequence of primes
>
>
>2, 3, 5, ..., p, .....
>
>
>Let me illustrate what I meant when I said primes were randomly distributed
>along the real line. Note when I said "randomly distributed" I don't mean they
>have a "uniform distribution". That is not what I mean when I say they are
>"randomly distributed." The exact nature of the distribution of prime numbers
>on the real line is a big area of research. Some mathematicians and quantum
>theorists think the random distribution of primes is related to quantum
>mechanics, quantum chaos and quantum computation.
>
>
>Suppose,
>
>
>P_{1}
>
>
>was the largest prime known to exist, discovered perhaps by a computer search
>using the sieve of Eratosthenes (a program using this would run very slow, I
>think!), or else maybe Pollard (p-1)--factorization or the Lucas-Lehmer test.
>
>
>Let
>
>
>P_{2} > P_{1}
>
>
>be an even much larger prime number, but such that P_{1} and P_{2} are NOT
>consecutive prime numbers. Suppose someone discovered the prime P_{2}, perhaps
>as I said with the Lucas-Lehmer test or using some other algorithm, and that
>this prime never was known to exist before.
>
>Suppose further that there still are unknown prime numbers existing between
>P_{1} and P_{2} > P_{1}?
>
>
>Question: Could Allen Poapst's algorithm find the sequence number for a
>recently discovered prime P_{2}, given that all the prime numbers between
>P_{1} and P_{2} on the real line still are not known?
>
>>
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>>End of Prime Digest, Vol 30, Issue 1
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