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Academic Paper from the year 2023 in the subject Mathematics - Analysis, grade: 2.00, , language: English, abstract: The work developed here brings a significant measure of order to the search for prime numbers.
It is shown conclusively that no Mersenne number can have an integer square root, this being, until now, one of the unsolved problems in Number theory. It is shown that there are a new set of numbers given by F = (M + 2)/3 where M is a Mersenne number. it is found that when M is prime then F is prime. however, it is also found that prime numbers may be generated when M is composite and the index of the Mersenne number is prime, but this is not invariably the case.
An accelerated method of trial division is derived and from which we construct one row matrices which we have called, templates. These are found to be of great utility in determining factors of numbers, and their 'extent' is equal to the number of digits in a number of interest. The template may be easily extended by a simple process explained in the text. Extension of a template by one unit increases that range of numbers which may be examined by an order of magnitude.