Infinite families of noncototients

• Autorzy: A. Flammenkamp , F. Luca
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 86
• Numer: 1
• Strony: 37-41

Daty

wydano

2000

otrzymano

1999-07-02

Twórcy

autor

A. Flammenkamp

• FSP Mathematisierung, Universität Bielefeld, Postfach 10 01 31, 33 501 Bielefeld, Germany

autor

F. Luca

• FSP Mathematisierung, Universität Bielefeld, Postfach 10 01 31, 33 501 Bielefeld, Germany

Bibliografia

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