In 2009, Heninger and Shacham presented an algorithm using the Hensel's lemma for reconstructing the prime factors of the modulus N = r1r2. This algorithm computes the prime factors of N in polynomial time, with high probability, assuming that a fraction greater than or equal to 59% random bits of its primes r1 and r2 is given. In this paper, we present the analysis of Hensel's lemma for a mult...

We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we ...

In this paper, we define prime (semiprime) hyperideals and prime(semiprime) fuzzy hyperideals of semihypergroups. We characterize semihypergroupsin terms of their prime (semiprime) hyperideals and prime (semiprime)fuzzyh yperideals.

A prime number is an integer bigger than l that has no factor except l and itself . A number that is not prime is called composite. The primality testing problem is to decide whether a given integer is prime or composite. It is considered to be well solved, in contrast to ine factoring problem, which asks for the factorization of a given integer into prime factors. As an example of what can be ...

Theorem 1.2. If all non-identity elements of G have the same order, this order is a prime p and |G| is a power of p. Proof. If |G| has two prime factors, say p and q, then G contains elements of orders p and q by Cauchy, which contradicts the hypothesis. Thus |G| has only one prime factor, say |G| = pm for a prime p. The orders of a non-identity element could be {p, p2, . . . , pm}. However, by...

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

Fast algorithms for computing the DHT of short transform lengths (N = 2, 3, 4, 5, 7, 8, 9 and 16) are derived. A new prime-factor algorithm is also proposed to compute the long-length DHTs from the short-length DHT algorithms. The short-length algorithms (except for N = 8 and N = 16) are such that the even and the odd parts of the DHT components are obtained directly, without any additional com...