Prime Numbers







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The technical paper containing Prime Number Distribution Series (PNDS) is now published1.

Download Material:

PNDS – Mathematical Formula for Finding Prime Numbers, by Yoldas Askan2.

PNDS – Sieve Verification C++ Open Source Code1.

PNDS Non-sieving Executable3.

Prime Number Distribution Series (PNDS), developed by the NSI team of mathematicians2, computes the exact count of primes below integer N (N → ∞) (π (N)). Aside from divide and count test and various sieving methods, it is believed that PNDS is the first and only exact mathematical formula for finding prime numbers.

PNDS demonstrates that a pattern of prime numbers exists; making fast evaluation of π (N) for large N at record speeds possible2.

Riemann Hypothesis

As shown in the technical paper, neither PNDS nor its inverse is a complex variable function, and that PNDS uniquely produces exact count of prime numbers for a given integer P, where 2 < P < ∞, it may be concluded that the distribution of prime numbers is not a complex variable problem and Riemann Zeta Function may not be pertinent to the subject matter.

History

Little or not at all considered fact is that the distribution of prime numbers can be regarded as piecewise orderly. For example, consider Euler prime formula, P(n) = n2 − n + 41. It is easy to show this function is asymptotic and strictly speaking only holds for 0 < n < 5. Like Euler prime formula, infinitely many series can be formulated each representing prime number distribution at the boundary limits. We therefore conclude that no single series, of any order; equation or a function will ever yield prime numbers for all n = 1 to exclusively.

Henceforward, Riemann conjecture that the nontrivial zeros of the Riemann zeta function all have real part ½ is an irrelevant submission to the problem.

However, a series can be formulated that will produce the boundary values of the piecewise orderly regions for N → ∞; as given by the PNDS mathematical formula

RSA Numbers, Public-key Cryptography and Online and Electronic Communication Security

PNDS series, together with other numerical algorithms, may allow factorization of large bit RSA numbers with expected computing times of the order of few minutes on a single processor.

Distribution List

Mathematical contents of the work have been disclosed to the following:
NSI Patent Attorney, Oct 2013
The Open University, May 2014
Published, August 2014
CodeProject, Sept 2014

1. Copyright © 2014 Nuclear Strategy, Inc. All rights reserved. Use of contents is restricted for educational and research purposes only. Do not reproduce, redistribute or reference without the prior written permission of the publisher.
2. Prime Number Distribution Series (PNDS) is discovered by Yoldas Askan
2. Current version of the PNDS exe computes π (1e12) in less than 10 seconds using a 2.49GHz Processor i5 Windows 8 workstation. However, PNDS is temporarily limited to N = 1e8 (evaluated in 3 seconds) pending high precision arithmetic library implementation on Windows (such as GNU GMP). PNDS will subsequently be unrestricted for any N → ∞. Please note that PNDS exe can potentially compute π (1e25) in less than 1 minute with similar specification Windows machines. Minimum requirements for running PNDS exe are Windows 7 or Windows 8 i5 machines.