Agrawali, Kayali ja Saxena teoreem algarvulisuse kohta

Abstract

This bachelor’s thesis gives an overview about prime numbers and different methods how to determine if an integer is prime or composite. It is based on Andrew Granville’s article "It is easy to determine if a given integer is prime", where he introduces and gives a proof of the primality theorem of Agrawal, Kayal and Saxena. The theorem was first published in 2002 in a paper "PRIMES is in P" by Manindra Agrawal, Neeraj Kayal and Nitin Saxena. This thesis consists of 3 parts. In the first part, the main definitions are given that are used throughout the whole thesis. The second part gives some examples about different theorems which have been used to determine primality before the theorem of Agrawal, Kayal and Saxena. In the third part we give a detailed proof of the main theorem of the thesis. It is formulated as follows. For a given integer n 2, let r be a positive integer < n, for which n has order > (log2 n)2 (mod r). Then n is prime if and only if 1) n is not a perfect power, 2) n does not have any prime factor r, 3) (x + a)n xn + a mod (n; xr =< 1) for each integer a, 1 a p r log n. Based on this theorem M. Agrawal, N. Kayal and N. Saxena created a deterministic primality-proving algorithm. This algorithm determines whether a positive integer n is prime or composite within polynomial time with respect to the number of digits of n.