Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n | ≤ 400 then |S | ≤ 32, and if |n |> 400 then |S | < 267.81 log |n | (log log |n|) 2. The question whether there exists an absolute bound (independent on n) for |S | still remains open. 1

1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1

Given f Z[x 1 , . . . , x n ], we compute the density of x such that f(x) is squarefree, assuming the abc conjecture. Given f, g Z[x 1 , . . . , x n ], we compute unconditionally the density of x such that gcd(f(x), g(x)) = 1. Function field analogues of both results are proved unconditionally. Finally, assuming the abc conjecture, given f Z[x], we estimate the size of the image of f({1, 2, . . . , n}) in Q # /Q #2 # {0}.

We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in particular, we obtain upper bounds for such twisted sums. 1

A square-free sieve is a result that gives an upper bound for how often a square-free polynomial may adopt values that are not square-free. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a square-free polynomial.

Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.

Abstract. We say that an integer n is k–free (k ≥ 2) if for every prime p the valuation vp(n) < k. If f: N → Z, we consider the enumerating function Sk f (x) defined as the number of positive integers n ≤ x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive integers up to x. We review the history of Sk f (x) in the special cases when f is the identity, the characteristic function of an arithmetic progression a polynomial, arithmetic. In each section we present the proof of the simplest case of the problem in question using exclusively elementary or standard techniques. 1. Introduction- The

The EKG or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n # 3, a(n) is the smallest natural number not already in the sequence with the property that gcd{a(n - 1), a(n)} > 1. In spite of its erratic local behavior, which when plotted resembles an electrocardiogram, its global behavior appears quite regular. We conjecture that almost all a(n) satisfy the asymptotic formula a(n) = n(1 + 1/(3 log n)) + o(n/ log n) as n # #; and that the exceptional values a(n) = p and a(n) = 3p, for p a prime, produce the spikes in the EKG sequence. We prove that {a(n) : n # 1} is a permutation of the natural numbers and that c 1 n # a(n) # c 2 n for constants c 1 , c 2 . There remains a large gap between what is conjectured and what is proved. 1.

[ILS], and Rubinstein [Ru], we use the 1- and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for small support the densities agree with Katz and Sarnak’s predictions. Further, the densities confirm that the curves ’ L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the 2-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry. 1.

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