Displaying similar documents to “Non-primes are recursively divisible.”

Landau’s problems on primes

János Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

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At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following : Are there infinitely many primes of the form n 2 + 1 ? ...

Infinite families of noncototients

A. Flammenkamp, F. Luca (2000)

Colloquium Mathematicae

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For any positive integer n let ϕ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ( 2 m k ) m ≥ 1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

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.

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

Hiroyuki Okazaki, Yasunari Shidama (2008)

Formalized Mathematics

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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...

On rough and smooth neighbors.

William D. Banks, Florian Luca, Igor E. Shparlinski (2007)

Revista Matemática Complutense

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We study the behavior of the arithmetic functions defined by F(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1) where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.