Sierpinski and Carmichael Numbers

Abstract

We establish several related results on Carmichael, Sierpinski and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2(exp n)k + 1 is not a Carmichael number for any n epilson N; this implies the existence of a set K of positive lower density such that for any k epsilon K the number 2(exp n)k + 1 is neither prime nor Carmichael for every K epilson N. Next, using a recent result of Matomaki we show that there are >> x1/5 Carmichael numbers up to x that are also Sierpinski and Riesel. Finally, we show that if 2(exp n)k+1 is Lehmer then n < 150 omega(k)2 log k, where omega(k) is the number of distinct primes dividing k.

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Document Details

Document Type
Technical Report
Publication Date
Jan 16, 2013
Accession Number
ADA621411

Entities

People

  • Carl Pomerance
  • Carrie Finch
  • Florian Luca
  • Pantelimon Stanica
  • William Banks

Organizations

  • Naval Postgraduate School

Tags

DTIC Thesaurus Topics

  • Applied Mathematics
  • Arithmetic
  • Composite Materials
  • Coverings
  • Equations
  • Inequalities
  • Information Operations
  • Intervals
  • Mathematics
  • Notation
  • Numbers
  • Prime Numbers
  • Rational Numbers
  • Schools
  • Three Dimensional
  • Universities

Fields of Study

  • Mathematics

Readers

  • Analytical Mechanics
  • Graph Algorithms and Convex Optimization.