On integers not of the form n - φ (n)

J. Browkin; A. Schinzel

ArticleOriginal scientific text

Title

On integers not of the form n - φ (n)

Authors ¹, ²

Affiliations

Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Mathematical Institute, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 137, 00-950 Warszawa, Poland

Abstract

W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers

2^{k} \cdot 509203

(k = 1, 2,...) is of the form n - φ(n).

A. Aigner, Folgen der Art $a r^{n} + b$ , welche nur teilbare Zahlen liefern, Math. Nachr. 23 (1961), 259-264.
P. Erdős, Über die Zahlen der Form σ(n)-n und n-φ(n), Elem. Math. 28 (1973), 83-86.
W. Keller, Woher kommen die größ ten derzeit bekannten Primzahlen?, Mitt. Math. Ges. Hamburg 12 (1991), 211-229.
W. Sierpiński, Number Theory, Part II, PWN, Warszawa, 1959 (in Polish).

Title

On integers not of the form n - φ (n)

Affiliations

Abstract

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