On Gaps between Squarefree Numbers Ii

On the Distribution of Gaps between Squarefree Numbers

Abc Allows Us to Count Squarefrees

We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erd} os: For any given polynomial f(x) 2 Zx], we deduce, from the abc-conjecture, an asymptotic estimate for the frequency with which f(n) is squarefree, when n is an integer (and also deduce such estimates for binary homogenous forms). Amongst several applications of this r...

-Catalan Numbers and Squarefree Binomial Coefficients

In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. We find all natural numbers n such that for p prime, p divides F (p, n), q ≥ 1 and all distinct residues of F (p, n) (mod p), q = 1, 2. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. We also prove that ( ...

p-Catalan Numbers and Squarefree Binomial Coefficients

In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p divides F (p, n), and also determine all distinct residues of F (p, n) (mod p), q ≥ 1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second...

Detecting squarefree numbers

We present an algorithm, based on the explicit formula for L-functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically and practically, and use it to prove that several RSA challenge numbers are not squarefull.