Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

**Yoshinori Fujisawa**- Shinshu University, Nagano
**Yasushi Fuwa**- Shinshu University, Nagano
**Hidetaka Shimizu**- Information Technology Research Institute, of Nagano Prefecture

- This article is concerned with Euler's theorem and small Fermat's theorem that play important roles in public-key cryptograms. In the first section, we present some selected theorems on integers. In the following section, we remake definitions about the finite sequence of natural, the function of natural times finite sequence of natural and $\pi$ of the finite sequence of natural. We also prove some basic theorems that concern these redefinitions. Next, we define the function of modulus for finite sequence of natural and some fundamental theorems about this function are proved. Finally, Euler's theorem and small Fermat's theorem are proved.

- Preliminary
- Finite Sequence of Naturals
- Modulus for Finite Sequence of Naturals
- Euler's Theorem and Small Fermat's Theorem

- [1]
Grzegorz Bancerek.
Cardinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
Joining of decorated trees.
*Journal of Formalized Mathematics*, 5, 1993. - [4]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Semigroup operations on finite subsets.
*Journal of Formalized Mathematics*, 2, 1990. - [7]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Yoshinori Fujisawa and Yasushi Fuwa.
The Euler's function.
*Journal of Formalized Mathematics*, 9, 1997. - [10]
Andrzej Kondracki.
The Chinese Remainder Theorem.
*Journal of Formalized Mathematics*, 9, 1997. - [11]
Jaroslaw Kotowicz.
Functions and finite sequences of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [12]
Rafal Kwiatek.
Factorial and Newton coefficients.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Rafal Kwiatek and Grzegorz Zwara.
The divisibility of integers and integer relatively primes.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [15]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [16]
Michal J. Trybulec.
Integers.
*Journal of Formalized Mathematics*, 2, 1990. - [17]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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