Factoring with cyclotomic polynomials

Factoring with Cyclotomic Polynomials * By Eric

This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials f(X) known to have the following property: given a multiple of f(p), we can quickly split any composite number that has p as a prime divisor. For example—taking f(X) to be X — 1—a multiple of p — 1 will suffice to easily factor any multiple of p, using an algorithm of Pollard....

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic polynomials with large coefficients

The coefficients a(m,n) and especially A(n) and S(n) have been the subject of numerous investigations (see [1] and the references given there). Until recently all these investigations concerned very thin sets of integers n. In [3] the author could establish a property valid for a set of integers of asymptotic density 1. Let ε(n) be any function defined for all positive integers such that limn→∞...

Cyclotomic Polynomials

Factoring Polynomials

Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent of computers and the resultant development of error-correcting codes, factoring over finite fields (e.g., Zp, for p a prime number) has become important as well. To understand this discussion, you need to know what polynomials are, and how to ...