Let φ(z) ∈ Q(z) be a rational function of degree d ≥ 2 with φ(0) = 0 and such that φ does not vanish to order d at 0. Let α ∈ Q have infinite orbit under iteration of φ and write φ(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, w...

Dedicated to Benjamin Fine on the occasion of his 60th birthday. The object of this paper is to give a proof of the following theorem: S/P ∼ = Λ mn ⊆ Q + , where S/P is a certain torsion-free factor group of the Baumslag-Solitar group a, b; a −1 b m a = b n | m = 0, n = 0, m, n ∈ Z, with m and n are relatively prime, and Λ mn is a subgroup of the additive group of the rational numbers Q + .

= −1 for all y satisfying y(y + 4)(y + 6)− 1 ≡ 0 (mod p). He confirmed that there are no such primes less than 109. In this paper, we describe the connection between the socialist primes and the left factorial function !n = 0! + 1! + · · · + (n − 1)! introduced by Ðuro Kurepa [4]. Kurepa conjectured that gcd(!n, n!) = 2 holds for all integers n > 1, which is equivalent to the statement that the...

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k−1)-dimensional skeleton and ( n−1 k ) facets such that Hk(X;Q) = 0. Here we introduce the following family of simplicial complexes. Let n, k be integers with k+1 and n relatively prime, and let A be a (k + 1)-element subset of the cyclic group Zn. The sum complex XA is the pure k-dimensional complex on the verte...

Abstract A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if $$a_{0,n} = p_n$$ a 0 , n = p denotes n th prime number $$a_{i,n} |a_{i-1,n}-...

We prove that the isomorphism class of the tame fundamental group of a smooth, connected curve over an algebraically closed eld k of characteristic p > 0 determines the genus g and the number n of punctures of the curve, unless (g, n) = (0, 0), (0, 1). Moreover, assuming g = 0, n > 1, and that k is the algebraic closure of the prime eld Fp, we prove that the isomorphism class of the tame fundam...

Let (bn) = (b1, b2, . . . ) be a sequence of integers. A primitive prime divisor of a term bk is a prime which divides bk but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial φ(z) is a sequence of integers (cn) where the n-th term is the n-th iterate of φ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that f...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...