New York Journal of Mathematics
Volume 28 (2022), 1152-1171


Abhishek Jha

On terms in a dynamical divisibility sequence having a fixed G.C.D with their indices

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Published: August 22, 2022.
Keywords: asymptotic density; Divisibility sequence; greatest common divisor; arithmetic dynamics; dynamical sequence; polynomial map.
Subject [2010]: Primary: 11C08. Secondary: 11A05, 11B05.

Let F and G be integer polynomials where F has degree at least 2. Define the sequence (an) by an=F(an-1) and a0=0. Let BF,G,k be the set of all positive integers n such that k | gcd(G(n),an) and if p | gcd(G(n),an) for some p, then p | k. Let AF,G,k be the subset of BF,G,k such that AF,G,k={n ≥ 1 : gcd(G(n),an)=k}. In this article, we prove that the asymptotic density of AF,G,k and BF,G,k exists for a class of (F,G) and also compute the explicit density of AF,G,k and BF,G,k for G(x)=x.


I would like to thank Emanuele Tron and Seoyoung Kim for looking at the article and providing valuable comments to improve its quality. I am thankful to Peter Mueller, David Speyer and Will Sawin for their answers on MathOverflow post [7] and Thomas Tucker for helpful discussions regarding Proposition 2.2 of the paper. I am grateful to Ayan Nath for his constant support and helpful advice. I am indebted to the anonymous referee for helpful comments.

Author information

Abhishek Jha:
Indraprastha Institute of Information Technology
New Delhi, India