Let [Formula: see text] be a finite group and fixed prime divisor of text]. We prove that if every maximal subgroup is nilpotent, or normal, has text]-order, then (1) solvable; (2) Sylow tower; (3) there exists at most one such neither text]-nilpotent nor text]-closed.

A long-standing difficulty for connectionism has been to implement compositionality, the idea of building a knowledge representation out of components such that the meaning arises from the meanings of the individual components and how they are combined. Here we show how a neural-learning algorithm, knowledge-based cascade-correlation (KBCC), creates a compositional representation of the prime-n...

Let φ(z) ∈ Q(z) be a rational function of degree d ≥ 2 with φ(0) = 0 and such that φ does not vanish to order d at 0. Let α ∈ Q have infinite orbit under iteration of φ and write φ(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, w...

Let g be an element of prime order p in an abelian group and α ∈ Zp. We show that if g, g, and g d are given for a positive divisor d of p−1, we can compute the secret α in O(log p·( √ p/d+ √ d)) group operations using O(max{ √ p/d, √ d}) memory. If g i (i = 0, 1, 2, . . . , d) are provided for a positive divisor d of p + 1, α can be computed in O(log p · ( √ p/d + d)) group operations using O(...

We study the function Θ(x, y, z) that counts the number of positive integers n ≤ x which have a divisor d > z with the property that p ≤ y for every prime p dividing d. We also indicate some cryptographic applications of our results.

Let g be an element of prime order p in an abelian group and α ∈ Zp. We show that if g, g, and g d are given for a positive divisor d of p − 1, we can compute the secret α in O(log p · ( p/d + √ d)) group operations using O(max{ p/d, √ d}) memory. If gαi (i = 0, 1, 2, . . . , d) are provided for a positive divisor d of p + 1, α can be computed in O(log p · ( p/d+ d)) group operations using O(ma...

1. Introduction. A prime q is a common index divisor of the algebraic number field Kii q divides the quotient d(u>)/d for all integers cu of K, where d(u>) is the discriminant of w and d is the discriminant of the field. A necessary condition is that q be less than the degree of

We give common generalizations of three formulae involving the number of relatively prime subsets of {1, 2, . . . , n} with some additional constraints. We also generalize a fourth formula concerning the Euler-type function Φk, and investigate certain related divisor-type, sum-of-divisors-type and gcd-sum-type functions.

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers. 2000 Mathematics Subject Classification. 11C20, 11D79, 11T23

Let Γ(Zm) be the zero divisor graph of the ring Zm. In this paper we explore the p-partite structures of Γ(Zm), as well as determine a complete classification of the chromatic number of Γ(Zm). In particular, we explore how these concepts are related to the prime factorization of m.