Let [Formula: see text] be a finitely generated field over and fix text]. We study the solutions of Catalan equation to solved in integers coprime with Our main result corrects earlier work Silverman.

The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k).

Let S be a finite set of positive integers. A “coprime base for S” means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P. There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best p...

The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k).

For each pair of coprime integers n > m ≥ 2 we construct pairs of non equivalent Conway irreducible hyperbolic knots with the same n-fold and m-fold cyclic branched covers. Mathematics Subject Classification (2000). Primary 57M25; Secondary 57M12, 57M50.

In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are pr...

We show that the Diophantine equation of the title has, for n > 1, no solution in coprime nonzero integers x, y and z. Our proof relies upon Frey curves and related results on the modularity of Galois representations.

1 . An arithmetic function f(n) is said to be additive if it satisfies the relation f(ob) = f(a)+f(b) for every pair of coprime positive integers a, b . In this paper we establish two results to the effect that an additive function which is not too large on many integers cannot often be large on the primes . If a l <a,< . . . is a sequence of positive integers, let A(x) denote ttie number of su...

In this paper, we propose a fast search-free method for direction-of-arrival (DOA) estimation with coprime arrays. A coprime array consists of two uniform linear subarrays with inter-element spacings Mλ/2 and Nλ/2, where M and N are coprime integers and λ is the wavelength of the signal. Because uniform linear arrays enjoy many processing advantages over arbitrary geometry arrays, our strategy ...

For any two consecutive Farey fractions γi = ai/qi < γi+1 = ai+1/qi+1, one has ai+1qi − aiqi+1 = 1 and qi + qi+1 > Q. Conversely, if q and q ′ are two coprime integers in {1, . . . , Q} with q + q > Q, then there are unique a ∈ {1, . . . , q} and a ∈ {1, . . . , q} for which aq − aq = 1, and a/q < a/q are consecutive Farey fractions of order Q. Therefore, the pairs of coprime integers (q, q) wi...