Primes Generated by Recurrence Sequences
Abstract:The notorious "Mersenne prime problem," which asks if infinitely many terms of the sequence 1,3,7,15,31,63,…. are prime, remains open. However, a closely related problem has a complete answer. Beyond the term 63, every number in the sequence has a primitive divisor (that is, a prime factor that does not divide any earlier term). We investigate the appearance of primitive divisors in sequences defined by quadratic polynomials, finding asymptotic estimates for the number of terms with primitive divisors. Along the way, we discuss how mathematicians use a mixture of heuristic and rigorous arguments to inform their expectations about prime appearance and primitive divisors in several natural recurrence sequences.
Document Type: Research Article
Publication date: 2007-05-01
More about this publication?
- American Mathematical Monthly publishes articles, notes, and other features about mathematics and the profession. AMM readers span a broad spectrum of mathematical interests, and include professional mathematicians as well as students of mathematics at all collegiate levels.
- Information for Authors
- Submit a Paper
- Subscribe to this Title
- Membership Information
- Information for Advertisers
- Terms & Conditions
- MAA Journals at ingentaconnect
- MAA Store
- Ingenta Connect is not responsible for the content or availability of external websites