Results 1  10
of
25
On the size of Diophantine mtuples
 Math. Proc. Cambridge Philos. Soc
"... 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 ..."
Abstract

Cited by 16 (13 self)
 Add to MetaCart
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
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 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 ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
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
Squarefree Values of Multivariable Polynomials
 Duke Math. J
, 2003
"... 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 un ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
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}.
On the Average Value of Divisor Sums in Arithmetic Progressions
, 2005
"... 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 multiplica ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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
ON THE SQUAREFREE SIEVE
, 2004
"... A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. 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 squarefree polynomial. ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. 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 squarefree polynomial.
Cyclicity of CM elliptic curves modulo p
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... 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. ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
Powerfree values, large deviations, and integer points on irrational curves
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2006
"... ..."
DENSER EGYPTIAN FRACTIONS
, 1998
"... An Egyptian fraction is a sum of reciprocals of distinct positive integers, so called because the ancient Egyptians represented rational numbers in that way. In an earlier paper, the author [8] showed that every positive rational number r has Egyptian fraction representations where the number of ter ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
An Egyptian fraction is a sum of reciprocals of distinct positive integers, so called because the ancient Egyptians represented rational numbers in that way. In an earlier paper, the author [8] showed that every positive rational number r has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator. More
Counting squarefree discriminants of trinomials under abc
 Proc. Amer. Math. Soc
"... Abstract. For an odd positive integer n ≥ 5, assuming the truth of the abc conjecture, we show that for a positive proportion of pairs (a, b) of integers the trinomials of the form t n + at + b (a, b ∈ Z) are irreducible and their discriminants are squarefree. 1. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. For an odd positive integer n ≥ 5, assuming the truth of the abc conjecture, we show that for a positive proportion of pairs (a, b) of integers the trinomials of the form t n + at + b (a, b ∈ Z) are irreducible and their discriminants are squarefree. 1.
1 and 2Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying
 Group Symmetries, Compositio Math
"... [ILS], and Rubinstein [Ru], we use the 1 and 2level densities to study the distribution of low lying zeros for oneparameter 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
[ILS], and Rubinstein [Ru], we use the 1 and 2level densities to study the distribution of low lying zeros for oneparameter 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 ’ Lfunctions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and SwinnertonDyer conjecture. By studying the 2level 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.