ACTA issues

Common divisors of the Euler function at related arguments

William D. Banks, Florian Luca, Igor E. Shparlinski

Acta Sci. Math. (Szeged) 72:3-4(2006), 525-536

Abstract. Let $\varphi $ denote the Euler function. For a fixed integer $k\not=0$, we study positive integers $n$ for which the largest prime factor of $\varphi(n)$ also divides $\varphi(n+k)$. We obtain an unconditional upper bound on the number of such integers $n\le x$, as well as unconditional lower bounds in each of the cases $k>0$ and $k< 0$. We also obtain some conditional lower bounds, for example, under the Prime $K$-tuplets Conjecture. Our lower bounds are based on explicit constructions.

AMS Subject Classification (1991): 11A25

Keyword(s): Euler function, largest prime factor, shift

Received July 6, 2004, and in revised form April 4, 2006. (Registered under 5934/2009.)