Displaying similar documents to “ Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ* ”

Properties of Primes and Multiplicative Group of a Field

Kenichi Arai, Hiroyuki Okazaki (2009)

Formalized Mathematics

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In the [16] has been proven that the multiplicative group Z/pZ* is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group.Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty...

On the Properties of the Möbius Function

Magdalena Jastrzebska, Adam Grabowski (2006)

Formalized Mathematics

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We formalized some basic properties of the Möbius function which is defined classically as [...] as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying many-sorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function.The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors.The formalization (which is...

Basic Properties of Primitive Root and Order Function

Na Ma, Xiquan Liang (2012)

Formalized Mathematics

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In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].

Pocklington's Theorem and Bertrand's Postulate

Marco Riccardi (2006)

Formalized Mathematics

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The first four sections of this article include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington's theorem (see [19]). The last section presents the formalization of Bertrand's postulate closely following the book [1], pp. 7-9.

The Perfect Number Theorem and Wilson's Theorem

Marco Riccardi (2009)

Formalized Mathematics

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This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson's theorem (that n is prime iff n > 1 and (n - 1)! ≅ -1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function Φ, proves that Φ is multiplicative and that Σk|n Φ(k) = n.