Infinite families of noncototients

A. Flammenkamp; F. Luca

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 37-41
  • ISSN: 0010-1354

Abstract

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For any positive integer n let ϕ(n) be the Euler function of n. A positive integer n is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression ( 2 m k ) m ≥ 1 consists entirely of noncototients. We then use computations to detect seven such positive integers k.

How to cite

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Flammenkamp, A., and Luca, F.. "Infinite families of noncototients." Colloquium Mathematicae 86.1 (2000): 37-41. <http://eudml.org/doc/210840>.

@article{Flammenkamp2000,
abstract = {For any positive integer $n$ let ϕ(n) be the Euler function of n. A positive integer $n$ is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression $(2^mk)_\{m ≥ 1\}$ consists entirely of noncototients. We then use computations to detect seven such positive integers k.},
author = {Flammenkamp, A., Luca, F.},
journal = {Colloquium Mathematicae},
keywords = {Euler's function; noncototient},
language = {eng},
number = {1},
pages = {37-41},
title = {Infinite families of noncototients},
url = {http://eudml.org/doc/210840},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Flammenkamp, A.
AU - Luca, F.
TI - Infinite families of noncototients
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 37
EP - 41
AB - For any positive integer $n$ let ϕ(n) be the Euler function of n. A positive integer $n$ is called a noncototient if the equation x-ϕ(x)=n has no solution x. In this note, we give a sufficient condition on a positive integer k such that the geometrical progression $(2^mk)_{m ≥ 1}$ consists entirely of noncototients. We then use computations to detect seven such positive integers k.
LA - eng
KW - Euler's function; noncototient
UR - http://eudml.org/doc/210840
ER -

References

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  1. [1] J. Browkin and A. Schinzel, On integers not of the form n-ϕ(n), Colloq. Math. 68 (1995), 55-58. Zbl0820.11003
  2. [2] P. Erdős, On integers of the form 2 k + p and related problems, Summa Brasil. Math. 2 (1950), 113-123. 
  3. [3] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1994. 
  4. [4] H. Riesel, Nοgra stora primtal [Some large primes], Elementa 39 (1956), 258-260 (in Swedish). 
  5. [5] W. Sierpiński, Sur un problème concernant les nombres k · 2 n + 1 , Elem. Math. 15 (1960), 73-74; Corrigendum, ibid. 17 (1962), 85. Zbl0093.04602
  6. [6] The Riesel Problem, http:/vamri.xray.ufl.edu/proths/rieselprob.html. 

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