Primes Generated by Recurrence Sequences

Authors: Everest, Graham; Stevens, Shaun; Tamsett, Duncan; Ward, Tom

Source: American Mathematical Monthly, Volume 114, Number 5, May 2007 , pp. 417-431(15)

Publisher: Mathematical Association of America

Key:
Free Content - Free Content
New Content - New Content
Subscribed Content - Subscribed Content
Free Trial Content - Free Trial Content

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

The full text electronic article is available for purchase. You will be able to download the full text electronic article after payment.

$20.00 plus tax      Refund Policy

 

OR

Back to top

Key:
Free Content - Free Content
New Content - New Content
Subscribed Content - Subscribed Content
Free Trial Content - Free Trial Content
Share this item with others: These icons link to social bookmarking sites where readers can share and discover new web pages.
Page Help Click here for Page Help
Shopping cart
Tools
Sign in






Need to register?
Sign up here
Text size: A | A | A | A