Главная

Jeffrey C. Lagarias

  1. On sets of integers which are both sum-free and product-free.

    Пнд, 01/09/2012 - 15:01 — SomNambul
    Authors: Jeffrey C. Lagarias, Par Kurlberg, Carl Pomerance
    Subjects: Number Theory
    Abstract

    We consider sets of positive integers containing no sum of two elements in
    the set and also no product of two elements. We show that the upper density of
    such a set is strictly smaller than 1/2 and that this is best possible.
    Further, we also find the maximal order for the density of such sets that are
    also periodic modulo some positive integer.

  2. Product-free sets with high density.

    Пт, 07/29/2011 - 15:01 — SomNambul
    Authors: Jeffrey C. Lagarias, Par Kurlberg, Carl Pomerance
    Subjects: Number Theory
    Abstract

    We show that there are sets of integers with asymptotic density arbitrarily
    close to 1 in which there is no solution to the equation ab=c, with a,b,c in
    the set. We also consider some natural generalizations, as well as a specific
    numerical example of a product-free set of integers with asymptotic density
    greater than 1/2.

  3. Smooth solutions to the equation A+B=C.

    Втр, 11/24/2009 - 16:01 — SomNambul
    Authors: K. Soundararajan, Jeffrey C. Lagarias
    Subjects: Number Theory
    Abstract

    This paper studies integer solutions to the ABC equation A+B+C=0 in which
    none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and
    set the smoothness S(A, B, C) to be the largest prime factor of ABC. We
    consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than
    a fixed power p of log H. Assuming the abc Conjecture we show that there are
    finitely many solutions if p<1. We discuss a conditional result, showing that
    the Generalized Riemann Hypothesis (GRH) implies there are infinitely many
    primitive solutions when p>8.

  4. Stochastic Models for the 3x+1 and 5x+1 Problems.

    Втр, 10/13/2009 - 18:46 — SomNambul
    Authors: Jeffrey C. Lagarias, Alex V. Kontorovich
    Subjects: Number Theory
    Abstract

    This paper discusses stochastic models for predicting the long-time behavior
    of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1
    problem. The stochastic models are rigorously analyzable, and yield heuristic
    predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.

  5. Stochastic Models for the 3x+1 and 5x+1 Problems.

    Втр, 10/13/2009 - 18:46 — SomNambul
    Authors: Jeffrey C. Lagarias, Alex V. Kontorovich
    Subjects: Number Theory
    Abstract

    This paper discusses stochastic models for predicting the long-time behavior
    of the trajectories of orbits of the 3x+1 problem and, for comparison, the 5x+1
    problem. The stochastic models are rigorously analyzable, and yield heuristic
    predictions (conjectures) for the behavior of 3x+1 orbits and 5x+1 orbits.

  6. The 3x+1 Problem: An Annotated Bibliography, II (2000-2009).

    Пнд, 08/31/2009 - 17:16 — SomNambul
    Authors: Jeffrey C. Lagarias
    Subjects: Number Theory
    Abstract

    The 3x+1 problem concerns iteration of the map T(n) =(3n+1)/2 if n odd; n/2
    if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the
    forward orbit of n includes the integer 1. This paper is an annotated
    bibliography of work done on the 3x+1 problem published from 2000 through 2009.
    This is a sequel to an annotated bibliography on the 3x+1 problem covering
    1963-1999.

    At present the 3x+1 Conjecture remains unsolved.

Math-arch

RSS-материал