Abstract
It is known, that any prime number has presentation in linear and quadratic forms. These properties may be used for finding class subsets of prime numbers. For that it is showed, that all prime numbers simple quadratic forms consist of a2+mb2, m=1,±2,3 . On these grounds it is examination for variants of prime numbers classification. It is discovered eight non-intersecting subsets of prime numbers, in conformity with equivalence classes modulo 24. The proposed classification is used for analyses Mersenne and Fermat numbers and composite numbers.
Highlights
The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of prime numbers
The difference between primes are come to light if investigate their linear and quadratic forms
That classification is natural and scientific, so it is at the same time the result and the important instrument of science research
Summary
The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of prime numbers. There are two subsets of primes with linear form 4n +1 and with linear form 4n + 3 It is practically simplest prime number classification with equivalence classes modulo 4. It is used for elementary analyzing and proving in Number Theory. The offered classification is based on the old researches, begun by Albert Girard (1595-1632) and Pierre de Fermat (1601-1665) and in detail analyzed in [1, §1.7]. It was found, that only the prime numbers of line series 4n +1 have a single presentation as a sum of two squares.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
