On sequences of integers with small prime factors
We show that the difference between consecutive terms in sequences of integers whose greatest prime factor grows slowly tends to infinity.
math.NT↗
arXiv subjects
C. L. Stewart
Publications and source records attributed to C. L. Stewart.
On prime factors of terms of binary recurrence sequences
We give estimates from below for the greatest prime factor of the n-th term of a binary recurrence sequence.
math.NT↗
Counting solvable $S$-unit equations
We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.
math.NT↗
On the representation of integers by binary forms
Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_F$ such that $R_F(Z)$ is asymptotic to $C_F Z^{\frac{2}{d}}$.
math.NT↗
On the representation of $k$-free integers by binary forms
Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$ and suppose that there is no prime $p$ such that $p^k$ divides $F(a,b)$ for all pairs of integers $(a,b)$. Let $R_{F,k}(Z)$ denote the number of $k$-free integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k} Z^{\frac{2}{d}}$ provided that $k$ exceeds $ \frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$.
math.NT↗
On divisors of Lucas and Lehmer numbers
Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdos from 1965.In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.
math.NT↗
Multivariate Diophantine equations with many solutions
We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in fewer than exp((4+o(1))s^{1/2}(log s)^{-1/2}) proper linear subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for m=2 [Compositio 36 (1988), 37-56]. Furthermore we prove that for any algebraic number field K of degree n, any integer m with 1<=m<n, and any sufficiently large s there are integers b_0,...,b_m in a number field which are linearly independent over the rationals, and prime numbers p_1,...,p_s, such that the norm polynomial equation |N_{K/Q}(b_0+b_1x_1+...+b_mx_m)|=p_1^{z_1}...p_s^{z_s} has at least exp{(1+o(1)){n/m}s^{m/n}(log s)^{-1+m/n}) solutions in integers x_1,..,x_m,z_1,..,z_s. This generalizes a result of Moree and Stewart [Indag. Math. 1 (1990), 465-472]. Our main tool, also established in this paper, is an effective lower bound for the number of ideals in a number field K of norm <=X composed of prime ideals which lie outside a given finite set of prime ideals T and which have norm <=Y. This generalizes a result of Canfield, Erdos and Pomerance [J. Number Th. 17 (1983), 1-28], and of Moree and Stewart (see above).
math.NT↗