In this article, we prove the existence and uniqueness of a certain distribution function on the unit interval. This distribution appears in Brent's model of the analysis of the binary gcd algorithm. The existence and uniqueness of such a function has been conjectured by Richard Brent in his original paper [1]. Donald Knuth also supposes its existence in [4] where developments of its properties...

1 SMP-based parallel algorithms and implementations for polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p-adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.

We consider a generalization of the gcd-sum function, and obtain its average order with a quasi-optimal error term. We also study the reciprocals of the gcd-sum and lcm-sum functions.

A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is the same. In particular, any family of trees has the common gcd property. Let F = {H1, . . . ,Hr} be a family of graphs having the common gcd property, and let d be the common gcd. It is proved that there exists a constant N = N(F ) such that for every n > N ...

A 3-equitable prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ..., |V |} such that if an edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)−f(v)) = 1, the label 2 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)− f(v)) = 2 and 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ b...

Richard Zippel’s sparse modular GCD algorithm is widely used to compute the monic greatest common divisor (GCD) of two multivariate polynomials over Z. In this report, we present how this algorithm can be modified to solve the GCD problem for polynomials over finite fields of small cardinality. When the GCD is not monic, Zippel’s algorithm cannot be applied unless the normalization problem is r...

We prove that if xm + axn permutes the prime field Fp, where m > n > 0 and a ∈ Fp, then gcd(m − n, p − 1) > √ p − 1. Conversely, we prove that if q ≥ 4 and m > n > 0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over Fq of the form xm + axn if and only if gcd(m,n, q − 1) = 1.

The atdvent of practical parallel processors has caused a reexamination of many existing algorithms with'the hope of discovering a parallel implementation. One of the oldest and best know algorithms is Euclid's algorithm for computing the greatest common divisor (GCD). In this paper we present a parallel algorithm to compute the GCD of two integers. Although there have been results in the paral...

A new parallel extended GCD algorithm is proposed. It matches the best existing parallel integer GCD algorithms of Sorenson and Chor and Goldreich, since it can be achieved in O (n/ logn) time using at most n1+ processors on CRCW PRAM. Sorenson and Chor and Goldreich both use a modular approach which consider the least significant bits. By contrast, our algorithm only deals with the leading bit...