#### Vol. 8, No. 2, 2019

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On a theorem of Hildebrand

### Carsten Dietzel

Vol. 8 (2019), No. 2, 189–191
##### Abstract

We give a short proof that for each multiplicative subgroup $H$ of finite index in ${ℚ}^{+}$, the set of integers $a$ with $a,a+1\in H$ is an IP-set. This generalizes a theorem of Hildebrand concerning completely multiplicative functions taking values in the $k$-th roots of unity.

##### Keywords
IP-set, multiplicative subgroup
Primary: 11B75
##### Milestones
Received: 29 January 2019
Revised: 7 February 2019
Accepted: 22 February 2019
Published: 20 May 2019
##### Authors
 Carsten Dietzel Institute of algebra and number theory University of Stuttgart Stuttgart Germany