Primes Generated by Recurrence Sequences

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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: May 1, 2007

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