Agrawali, Kayali ja Saxena teoreem algarvulisuse kohta
Date
2013
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Tartu Ülikool
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.
Description
Keywords
algarv, Agrawali, Kayali ja Saxena teoreem, algoritm, Manindra Agrawal, Neeraj Kayal, Nitin Saxena, Andrew Granville, bakalaureusetööd