Selected Chapters of Number Theory: Special Numbers
1 primary work • 2 total works
Book 0
This book contains a complete detailed description of two classes of special numbers closely related to classical problems of the Theory of Primes. There is also extensive discussions of applied issues related to Cryptography.In Mathematics, a Mersenne number (named after Marin Mersenne, who studied them in the early 17-th century) is a number of the form Mn = 2n - 1 for positive integer n.In Mathematics, a Fermat number (named after Pierre de Fermat who first studied them) is a positive integer of the form Fn = 2k+ 1, k=2n, where n is a non-negative integer.Mersenne and Fermat numbers have many other interesting properties. Long and rich history, many arithmetic connections (with perfect numbers, with construction of regular polygons etc.), numerous modern applications, long list of open problems allow us to provide a broad perspective of the Theory of these two classes of special numbers, that can be useful and interesting for both professionals and the general audience.
Book 2
Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers) and give much of their properties, facts and theorems with full proofs of them.In the book, a complete detailed description of two classes of special numbers, perfect and amicable numbers, as well as their numerous analogue and generalizations, is given. Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians.This is also an important part of the history of prime numbers, since the main formulas generated perfect and amicable pairs, depends on the good choice of one or several primes of special form.Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems, as well as a lot of important computer algorithms needed for searching for new large elements of these two famous classes of special numbers. The mathematical part of this theory is closely connected with classical Arithmetics and Number Theory. It contains information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogue (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc.), their connections with other classes of special numbers, etc.Moreover, perfect and amicable numbers are involved in the search for new large primes, and have numerous connections with contemporary Cryptography. For these applications, one should study well-known deterministic and probabilistic primality tests, standard algorithms of integer factorization, the questions and open problems of Computational Complexity Theory.