MATHEMATICS

The papers in this category will present some new ideas on the generation of prime numbers, their distribution among the natural numbers and for primality testing. Analgorithmic version of Eratosthenes Sieve together with additional the determination of the prime factors of large composites. The stabilisation of the Bairstow method of determining the roots of high order polynomial equations is addressed, and a new method for the precise determination of the multiple roots of such polynomials. A simple proof of Fermat's Last Theorm is also described and a resolution of the Riemann Hypothesis is also presented. The links below are to the papers in PDF format.

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, BairstowMethod3.XLS and PolynomialContruction1.2.0.XLS, both of which can be downloaded here as a ZIP file, (These include implementation of M4 and M5 below), Bairstow3.zip.

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, BairstowMethod3.XLS, together with a Polynomial Construction Spreadsheet, (PolynomialConstruction1.2.0.xls), both of which can be downloaded here as a ZIP file, (These include implementation of M5 below)Bairstow3.zip.

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 Polynomial Construction Spreadsheet, (PolynomialConstruction1.2.0.xls), both of which can be downloaded here as a ZIP file, Bairstow3.zip.

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.Prime Generation Spreadsheet.

Primality Testing - A New Method. (M7 Version 2.0.0) - This paper presents a new method for the primality testing of the natural numbers. This method is based upon an attribute of the Natural numbers designated, Multiples. For any Natural number N, Multiples is the sum of the integer results of dividing N by all the prime numbers less than the square root of N. Any Natural number is shown to be prime if its Multiples have the same value as its predecessor. The main advantage of this method over all others is that, if implemented appropriately, it provides a virtually instantaneous result. Mitigated against this is that a prime number database is required, and a considerable amount of coding is necessary to achieve it. Two demonstration EXCEL spreadsheets are available with this paper, which includes the necessary prime number database. The first and smaller, is as described in the paper. The second is an extended version which will test numbers up to 2.5E13 for primality in less than 1 second, (depending upon processor speed).
Basic Primality Test Spreadsheet.
Extended Primality Test Spreadsheet.

The Distribution of Primes Among the Natural Numbers. (M8 Version 1.0.0) - This paper presents a new method for determining the distribution of primes among the Natural numbers. It is a counting function which uses the Multiples definition of a prime number, as shown in M7 above, to determines the number of primes within a range of any two limits, the lower of which can be unity. It also enables a single number to be tested for primality, although this is best accomplished using M7 above. Associated with the paper, is a demonstration macro driven EXCEL spreadsheet. This incorporates a prime number database used by the spreadsheet during operation, and which can be updated by the spreadsheet itself as calculation progresses to extend its capabilities. The updated database can then also be transferred the spreadsheets associated with paper M7, to increase their databases, and thereby extend their capabilities in the primality testing of ever larger numbers.
Prime Distribution Test Spreadsheet.

The Riemann Hypothesis - A Resolution. (M9 Version 1.0.0) - The first half of this paper presents a resolution of the Riemann Hypothesis by converting the Riemann Zeta Function into functions of Sine and Cosine of the argument, and then exploring its value for selected values of the exponenent, s. The result is that the Zeta function is seen to exhibit oscillatory characteristics which determine its behaviour as the argument is increased. The overall result from this approach is that the Riemann Hypothesis is determined to be false. This result is an unexpected one, and consequently past analytical methods used to investigate this hypothesis are reviewed in the second half of the paper for their veracity. Finally, a review of the mathematical methods to treat infinite series such as the Riemann Zeta Function are also reviewed.

An Algorithmic Version of the Sieve of Eratosthenes. (M19 Version 1.0.0) - This paper presents the development of an algorithmic version of the Sieve of Eratosthenes, to simplify the identification of prime numbers. The method also provides for the calculation of p(N), the number of primes from zero to some number N. In addition, it also provides a new and very simple means to test any number for primality. Finally, where a number is known to be composite, the method provides for a very simple procedure for determining its factors.

The Closed Forms of Convergent Infinite Series. (M10 to M18)- This section contains a number of papers concerning the determination of the closed forms of convergent infinite series. Several are based upon the work of Leonhard Euler in the eighteeenth century, while others detail new methods, generalisation and series that are dependent upon logarithmic functions.