Prime Numbers : - (M1 Ver 1.0.0). This paper discusses the generation of prime numbers by single variable polynomial equations of order 2 or higher. The types considered are controlled by a strict set of criteria which limits those acceptable. Subsequently, applicable equations are categorized into four groups only one of which is acceptable under the control criteria. A conversion process is then described which allows non-conforming equations to be converted to a version which is. Finally the complete set of equations required to generate all primes between 0 and 100 is presented, followed by a challenge for readers to find the remaining equations required to generate all the primes between 101 and 200.

Prime Numbers 2: - (M6 Ver 1.0.0). This paper continues the investigation into the generation of prime number sequences by quadratic equations, with the presentation of two search methods. The first, the Composite Search method, searches for the lowest composite within the range of a possible prime generating quadratic, and determines the number of primes in the sequence generated that is lower than this composite. The second method is a re-configurable version of Ulam's Spiral, that enables prime generating quadratics to be found with any value of quadratic coefficients. These two methods have in combination enabled quadratics to be found that generate all the prime numbers between zero and 1500, plus nearly 200 others above this range. Both methods are provided as EXCEL spreadsheets in the form of a downloadable ZIP file.

Fermat's Last Theorem - A Simple Proof : - (M2 Ver 1.0.0). Fermat's Last Theorem, (or Conjecture), is probably the most famous mathematical problem in the history of the subject and remained unproven for over 360 years. It was finally solved in 1995 via a very complex analysis using advanced modern techniques in pure mathematics. The simple proof provided here uses nothing more than the same level of analysis that would have been available in Fermat's day, the mid-seventeenth century.

Bairstow Polynomial Roots: - (M3 Ver 1.0.0). This paper discusses the Bairstow Method of finding the roots of high order polynomial equations. The frequent tendency of this method to diverge, or hunt, is analysed and procedures to avoid these problems developed. A process for using the Bairstow Method to find the roots of polynomials with complex co-efficients is also presented. In addition, several new algorithms for the accurate extraction of multiple identical roots, are also described. The results of this excercise are implemented in two experimental spreadsheets, Bairstow.XLS and Polynomial Contruction.XLS, both of which can be downloaded here as a ZIP file, Bairstow3.zip, (Final Version as discussed below).

Polynomial Multiple Root Extraction - 1 - The DDR Method: - (M4 Ver 1.0.0). This paper discusses a new method for the precise determination of the roots of high order polynomials, when those roots occur in multiple pairs. The method is termed the Differential Division Remainder Method, and is based upon the fact that when poylynomials in this category are differentiated with respect to their independent variable, and the differential is then divided into the original equation, the ratios of the first three remainder terms are quadratic functions of the primary root. The method is very easy to apply manually, which is described and has here also been implemented in macro driven EXCEL spreadsheet form, Bairstow Method2.XLS, together with a slightly updated Polynomial Construction Spreadsheet, (POLYNOMIALCONSTRUCTION1.1.0.XLS), both of which can be downloaded here as a ZIP file, Bairstow3.zip, (Final Version as discussed below).

Polynomial Multiple Root Extraction - 2 - The CDA Method: - (M5 Ver 1.0.0). This paper discusses a second method for the precise determination of the multiple plus singular roots of high order polynomials, when those roots occur in any combination. The method is termed the Cascade Differential Analysis Method, and is based upon the fact that when polynomials in this category contain p multiple roots, when the polynomial is differentiated (p - 1) times, the reduced polynomial contains one of the multiple roots plus others reflecting the dynamics of the original equation. The multiple root can then be confirmed and extracted from the original equation. The method hes been implemented here in macro driven EXCEL spreadsheet form, Bairstow Method3.XLS, together with a slightly updated Polynomial Construction Spreadsheet, (POLYNOMIALCONSTRUCTION1.2.0.XLS), both of which can be downloaded here as a ZIP file, Bairstow3.zip.