12,778 research outputs found
Arithmetic properties of blocks of consecutive integers
This paper provides a survey of results on the greatest prime factor, the
number of distinct prime factors, the greatest squarefree factor and the
greatest m-th powerfree part of a block of consecutive integers, both without
any assumption and under assumption of the abc-conjecture. Finally we prove
that the explicit abc-conjecture implies the Erd\H{o}s-Woods conjecture for
each k>2.Comment: A slightly corrected and extended version of a paper which will
appear in January 2017 in the book From Arithmetic to Zeta-functions
published by Springe
An Erd\H{o}s-Kac theorem for integers with dense divisors
We show that for large integers , whose ratios of consecutive divisors are
bounded above by an arbitrary constant, the number of prime factors follows an
approximate normal distribution, with mean and variance , where and . This result
is then generalized in two different directions.Comment: 28 page
Neighboring ternary cyclotomic coefficients differ by at most one
A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and
r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest
ones for which the behaviour of the coefficients is not completely understood.
Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients
differ by at most four. We show that, in fact, they differ by at most one.
Consequently, the set of coefficients occurring in a ternary cyclotomic
polynomial consists of consecutive integers.
As an application we reprove in a simpler way a result of Bachman from 2004
on ternary cyclotomic polynomials with an optimally large set of coefficients.Comment: 11 pages, 2 table
A coprimality condition on consecutive values of polynomials
Let be quadratic or cubic polynomial. We prove that there
exists an integer such that for every integer one can
find infinitely many integers with the property that none of
is coprime to all the others. This extends
previous results on linear polynomials and, in particular, on consecutive
integers
Arithmetic Progressions in a Unique Factorization Domain
Pillai showed that any sequence of consecutive integers with at most 16 terms
possesses one term that is relatively prime to all the others. We give a new
proof of a slight generalization of this result to arithmetic progressions of
integers and further extend it to arithmetic progressions in unique
factorization domains of characteristic zero.Comment: Version 2 (to appear in Acta Arithmetica) with minor typos correcte
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