A study is presented of right (left) coprime decompositions of a collection of N -periodic rational matrices, with some ordered structure. From a block-ordered right coprime decomposition of a rational matrix of the given periodic collection, the corresponding block-ordered right coprime decompositions of the remaining matrices of the collection are constructed. In addition, those decomposition...

Let α be an algebraic number of degree d > 2 over Q. Suppose for some pairwise coprime positive integers n1, . . . , nr we have deg(αj ) < d for j = 1, . . . , r, where deg(α) = d for each positive proper divisor n of nj . We prove that then φ(n1 . . . nr) 6 d, where φ stands for the Euler totient function. In particular, if nj = pj , j = 1, . . . , r, are any r distinct primes satisfying deg(α...

Given two integers q and k, for any prime r not dividing q with r ≡ 1 mod k, we denote by indr(q) the index of q mod r. In [2] the question was raised of calculating the density of the primes r for which indr(q) and (r− 1)/k are coprime; this is the condition that the Gauß period in Fq(r−1)/k defined by these data be normal over Fq. We assume the Generalized Riemann Hypothesis and calculate a f...

Let (X , x0) be any one–pointed compact connected Riemann surface of genus g, with g ≥ 3. Fix two mutually coprime integers r > 1 and d. Let MX denote the moduli space parametrizing all logarithmic SL(r, C)–connections, singular over x0, on vector bundles over X of degree d. We prove that the isomorphism class of the variety MX determines the Riemann surface X uniquely up to an isomorphism, alt...

For any integer n ≥ 2 and any nonnegative integers r, s with r + 2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r, s), Galois group Sn, and squarefree discriminant; we may also force the discrimina...

Let A be a pre-defined set of rational numbers. We say a set of natural numbers S is an A-quotient-free set if no ratio of two elements in S belongs to A. We find the maximal asymptotic density and the maximal upper asymptotic density of A-quotient-free sets when A belongs to a particular class. It is known that in the case A = {p, q}, where p, q are coprime integers greater than one, the lates...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K of degree d, and denote its Jacobian by J . Denote the Mordell–Weil rank of J(K) by r. We give an explicit and practical Chabauty-style criterion for showing that a given subset K ⊆ C(K) is in fact equal to C(K). This criterion is likely to be successful if r ≤ d(g − 1). We also show that the only sol...

If F (x, y) ∈ Z[x, y] is an irreducible binary form of degree k ≥ 3 then a theorem of Darmon and Granville implies that the generalized superelliptic equation F (x, y) = z has, given an integer l ≥ max{2, 7 − k}, at most finitely many solutions in coprime integers x, y and z. In this paper, for large classes of forms of degree k = 3, 4, 6 and 12 (including, heuristically, “most” cubic forms), w...

Consider a finite Abelian group (G,+), with |G| = p , p a prime number, and φ : G → G the cellular automaton given by (φx)n = μxn + νxn+1 for any n ∈ N, where μ and ν are integers coprime to p. We prove that if P is a translation invariant probability measure onG determining a chain with complete connections and summable decay of correlations, then for any w = (wi : i < 0) the Cesàro mean distr...

Let Cn be the cyclic group of n elements, and let S = (a1, · · · , ak) be a sequence of elements in Cn. We say that S is a zero sequence if ∑k i=1 ai = 0 and that S is a minimal zero-sequence if S is a zero sequence and S contains no proper zero subsequence. In this paper we prove, among other results, that if S is a minimal zero sequence of elements in Cn and |S| ≥ n − [ 3 ] + 1, then there ex...