PRIME NUMBER GENERATING QUADRATICS - 2 - COMPOSITE SEARCH AND ULAM'S SPIRAL, (EXTENDED).

The links below are to the Sections of the paper on two methods of searching for quadratic equations that generate continuous sequences of prime numbers. The methods described have enabled the identification of some 450 quadratics, within which just 104 generate all of the prime numbers between zero and 1500, plus nearly 200 others above this range.

Introduction - The first section contains a title page, a brief abstract, and the main introduction.

Prime Number Quadratic Generation - The second section provides a detailed description of both the semi-analytic Composite Search, and the re-configurable Ulam's Spiral methods of identifing prime number sequence generating quadratic equations. The theory behind the manner in which these operate is fully described.

Computerisation - The third Section describes the macro driven EXCEL spreadsheets that provide computerisation of the above two methods. Details of how these can be used to identify more such quadratics is included. Both spreadsheets are made available as a downloadable ZIP file in this Section.

Conclusions and Appendix A - Concluding Remarks are presented here together with the first Appendix. The latter presents a theoretical derivation of the quadratic co-efficients that the re-configurable Ulam's Spiral generates from the sequence terms. The derivation is primarily based upon the variability of the Base Number, (the number at the heart of the spiral), spiral Step Size, and the number of spiral boundary crossings effected by the sequence terms.

Appendix B - The second Appendix provides examples of the manner in which the spiral can be re-configured to line up a complete sequence of quadratically generated primes in a single line in the spiral.

Appendix C and References - The final Appendix presents the maximisation of some well known prime sequence generating quadratics and pertinent comments thereto.

Return to the Start Page for this Category:- Mathematics.

Return to the home page for this Site:- Home.