Terms in elliptic divisibility sequences divisible by their indices

TL;DR

This paper investigates the indices n for which n divides the n-th term of an elliptic divisibility sequence, providing methods to construct new such indices and bounds for exceptional cases.

Contribution

It introduces a construction method for indices in S(D) based on prime divisors and aliquot cycles, and establishes bounds for exceptional indices not generated by these methods.

Findings

Construction of new indices in S(D) from prime divisors and aliquot cycles

Bounds established for exceptional indices outside the construction methods

Insights into the structure of indices dividing their corresponding sequence terms

Abstract

Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D), we explain how to construct elements nd in S(D), where d is either a prime divisor of D_n, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.