# Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*

Formalized Mathematics (2008)

• Volume: 16, Issue: 2, page 103-107
• ISSN: 1426-2630

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## Abstract

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In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if Z/pZ is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ* = Z/pZ{0is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/pZ*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ* and prove it is cyclic.MML identifier: INT 7, version: 7.8.10 4.99.1005

## How to cite

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Hiroyuki Okazaki, and Yasunari Shidama. " Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ* ." Formalized Mathematics 16.2 (2008): 103-107. <http://eudml.org/doc/266839>.

@article{HiroyukiOkazaki2008,
abstract = {In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if Z/pZ is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ* = Z/pZ\{0is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/pZ*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ* and prove it is cyclic.MML identifier: INT 7, version: 7.8.10 4.99.1005 },
author = {Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {2},
pages = {103-107},
title = { Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ* },
url = {http://eudml.org/doc/266839},
volume = {16},
year = {2008},
}

TY - JOUR
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*
JO - Formalized Mathematics
PY - 2008
VL - 16
IS - 2
SP - 103
EP - 107
AB - In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if Z/pZ is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ* = Z/pZ{0is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/pZ*. Moreover we prove the uniqueness of factoring an integer. Next we define the multiplicative group Z/pZ* and prove it is cyclic.MML identifier: INT 7, version: 7.8.10 4.99.1005
LA - eng
UR - http://eudml.org/doc/266839
ER -

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