Recent developments in primality testing

Abstract

Is there an algorithm that can decide whethern is prime or composite and that runs in polynomial time? The Adleman-Rumely algorithm and the Lenstra variations come so close, that it would seem that almost any improvement would give the final breakthrough. Is factoring inherently more difficult than distinguishing between primes and composites? Most people feel that this is so, but perhaps this problem too will soon yield. In his “Disquisitiones Arithmeticae” Gauss [9], p. 396, wrote “The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimable men, i.e., for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers.” The struggle continues!