2.5  The Challenge and the PRIMES Program.

It can be seen from Table 2.1 and the previous Section, that not only have the first 26 prime numbers between 0 and 100 been generated by Eqs. (2.5) to (2.11), but also the majority of those between 101 and 200. Those not covered in this latter range are:- 101, 103, 137, 163, 167, 179, 181 and 193.

Finding the equations, which, while conforming to the criteria set in Section 2.1, will generate the above numbers, (as well as repeats and others), is left as a challenge for any interested reader. Equations so found should be forwarded via the email address on the LINKS page of this website, and will be included in the next update of this article, together with an acknowledgement.

To assist in this task, a Visual Basic program, PRIMES, can be downloaded and used to check the numbers generated by any potential equation for primeness. The program will check any single number for primeness, (up to a maximum of 2,147,483,647), and/or generate all the prime numbers within any range of 9,000 up to the same limit. The PRIMES program can be downloaded from the website introduction page for this article, or here - (Primes.zip ).

In participating in this exercise, apart from the criteria, there are a some significant points to note.

(i) There are many equations which generate prime numbers which duplicate those already covered above. Some of these may also generate new numbers albeit not those that are the subject of this challenge. Such equations should also be forwarded for inclusion in the next or subsequent update.

(ii) Only quadratic equations should be considered. Higher orders will be the subject of future updates.

(iii) While a manual process is quite feasible in conducting this exercise, it would be both very laborious and tedious, and therefore for those who wish to participate it would be preferable to adopt some form of computer analysis.

(iv) It is already known that there are no acceptable equations with a greater a0 than in (2.5) and (2.8) for those particular types of equations. It is not known whether this limit applies to all quadratic equations constructed for this purpose. However, note that the higher the value of a0, the more primes will need to be generated in order for the criteria to be met.

(v) There are at most seven equations needed to generate the above eight primes.

3  A Simple Method of Equation Generation.

This spreadsheet based computerised methodology will be included in the next update to this article.

4  Conclusions.

The criteria that have been adopted here severely restricts the number of acceptable equations. However, it does ensure that all equations that conform, present a highly standardised set of characteristics. Also, working to cover all prime numbers from zero upwards, provides a logical and systematic approach that may subsequently result in equation patterns that will aid in further identification.

The goal that has been set in this initial paper is a very limited one, but it is only a start. It is not known whether polynomial equations of the type utilised here can eventually generate prime numbers ad infinitum, but as progress materialises it is hoped that future targets will become more ambitious and challenging.



M1 Version 1.0.0
Ó P.G.Bass, March 2009
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