It is a classical result that the probability two positive integers n, m ≤ x are relatively prime tends to 1/ζ(2) = 6/π2 as → ∞. In this paper, same still true when n and restricted sub-sequences, i.e. Piatetski–Shapiro sequence, Beatty sequence floor function set.

An eecient algorithm is developed for determining the greatest common left divisor (GCLD) of two polynomial matrices. Knowing this divisor allows for several immediate applications: In coding theory, a noncatastrophic convolutional encoder can be derived from an arbitrary one. In systems theory, irreducible matrix fraction descriptions of transfer function matrices can be found. In linear algeb...

Greatest common divisor algorithms are used to provide a natural motivation for considering a class of Jacobi-Perron algorithms which includes the original Jacobi algorithm. This work proves convergence and establishes metric properties for one of these algorithms. The proofs generalize to the larger class of algorithms. Full connections with the calculation of greatest common divisors will be ...

Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, P (Np) > p, where θ(p) is any function of p such that θ(p)→ 0 as p→∞. Moreover ...

Greatest common divisors and least common multiples of quotients of elements of integral domains have been investigated by Lüneburg and further by Jäger. In this paper we extend these results to invertible fractional ideals. We also lift our earlier study of the greatest common divisor and least common multiple of finitely generated faithful multiplication ideals to finitely generated projectiv...