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.

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, Bairstow.zip

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