Abstract
With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number N > 1 N > 1 is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practical tests for primality have recently been given (running in deterministic linear time) in terms of pseudoprimes for certain second- or third-order linear recurrence sequences. Here, a powerful general theory is described to characterize pseudoprimes for higher-order recurrence sequences. This characterization leads to a broadening and strengthening of practical primality tests based on such pseudoprimes.
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