Vassiliev invariants up to order six for arbitrary torus knots {n,m}, with n and m coprime integers, are computed. These invariants are polynomials in n and m whose degree coincide with their order. Furthermore, they turn out to be integer-valued in a normalization previously proposed by the authors. ⋆ e-mail: [email protected]

For three positive integers ai, aj , ak pairwise coprime, we present an algorithm that find the least multiple of ai that is a positive linear combination of aj , ak. The average running time of this algorithm is O(1). Using this algorithm and the chinese remainder theorem leads to a direct computation of the Frobenius number f(a1, a2, a3).

In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q.

Here, we show that if s �∈ {1, 2, 4} is a fixed positive integer and m and n are coprime positive integers such that the multiplicative order of Fn+1/Fn modulo Fm is s, where Fk is the kth Fibonacci number, then m < 500s2.

For the given coprime polynomials over integers, we change their coefficients slightly over integers so that they have a greatest common divisor (GCD) over integers. That is an approximate polynomial GCD over integers. There are only two algorithms known for this problem. One is based on an algorithm for approximate integer GCDs. The other is based on the well-known subresultant mapping and the...

A. We use the group (Z2,+) and two associated homomorphisms, τ0, τ1, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote T . While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the paren...

The factor refinement principle turns a partial factorization of integers (or polynomials) into a more complete factorization represented by basis elements and exponents, where basis elements are pairwise coprime. There are many applications of this refinement technique such as simplifying polynomial systems and, more generally, to algebraic algorithms generating redundant expressions during in...

Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m > 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

We give a new proof of the validity of Cornacchia’s algorithm for finding the primitive solutions (u, v) of the diophantine equation u + dv = m, where d and m are two coprime integers. This proof relies on diophantine approximation and an algorithmic solution of Thue’s problem.

For three positive integers ai, aj , ak pairwise coprime, we present an algorithm that find the least multiple of ai that is a positive linear combination of aj , ak. The average running time of this algorithm is O(Loga1). Using this algorithm and the chinese remainder theorem leads to a direct computation of the Frobenius number f(a1, a2, a3).