The terms in Lucas sequences divisible by their indices

TL;DR

This paper characterizes the indices n for which n divides terms in Lucas sequences of the first and second kind, revealing their structure as products of specific basic numbers and primes, with detailed properties of these primes.

Contribution

It provides a complete description of the indices dividing Lucas sequence terms, extending previous work by explicitly identifying the prime factors involved.

Findings

Indices n dividing u_n or v_n are products of 1, 6, or 12 and specific primes.

The set of such primes has particular properties detailed in the paper.

The structure of these indices is fully characterized for both sequence types.

Abstract

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for which n divides u_n and also the set of indices n for which n divides v_n. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called basic number, which can only be 1, 6 or 12, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.