Extended Euclidean Algorithm and CRT Algorithm

Open access


In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based on the source code of the NZMATH, a number theory oriented calculation system developed by Tokyo Metropolitan University [8].

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

[4] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

[5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[7] Czesław Bylinski. The sum and product of finite sequences of real numbers. FormalizedMathematics, 1(4):661-668, 1990.

[8] NZMATH development Group. http://tnt.math.se.tmu.ac.jp/nzmath/.

[9] Donald E. Knuth. Art of Computer Programming. Volume 2: Seminumerical Algorithms, 3rd Edition, Addison-Wesley Professional, 1997.

[10] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.

[11] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.

[12] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

[13] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

Formalized Mathematics

(a computer assisted approach)

Journal Information

SCImago Journal Rank (SJR) 2017: 0.119
Source Normalized Impact per Paper (SNIP) 2017: 0.237

Target Group

researchers in the fields of formal methods and computer-checked mathematics


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 126 126 14
PDF Downloads 37 37 5