Primality tests: an infinity of exceptions.
So-called Carmichael numbers rank at the top of the sneakiness chart. Though rare, they can confound tests based on Pierre de Fermat's "little theorem," passing for prime numbers -- those divisible only by themselves and one -- when they really represent several smaller numbers multiplied together.
Now a trio of mathematicians at the University of Georgia in Athens has proved there are infinitely many Carmichael numbers. Their proof settles an issue that dates back to 1910, when mathematician R.D. Carmichael first uncovered such numbers, computed 15 examples, and stated without proof that "this list might be indefinitely extended."
The new proof, which represents the work of William R. (Red) Alford, Andrew Granville, and Carl Pomerance, also highlights the intrinsic inadequacy of certain types of tests employed within various widely used commercial computer programs for rapidly verifying that a given number is prime. Knowing whether a number is prime plays a key role in several cryptographic schemes for assuring computer security (SN: 9/7/91, p.148).
"The result itself is not surprising," says mathematician Hugh C. Williams of the University of Manitoba in Winnipeg. "What's interesting is that now there is a proof, and the proof is actually quite short, very elegant, very clever."
Granville describes the steps leading to the proof in the September NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY.
The simplest way of determining primality is by trial division -- dividing the given number by every number between 2 and the square root of the given number. If these trial divisors all leave a remainder, then the given number must be a prime. That's easy to do with small numbers, such as 107 or 11,035, but impossibly time-consuming for, say, 100-digit numbers.
In 1640, Fermat noticed a certain arithmetical relationship that pointed to a potential shortcut for culling lists of numbers. He discovered that if a number n is prime, then n divides evenly into [b.sup.n] - b. Thus, for n = 11 and b = 2, [2.sup.11] - 2 = 2,046, and 11 divides evenly into 2,046. This works for all prime numbers.
If there is a remainder, the given number is composite. However, a few composite numbers also leave no remainder for one or more choices of b, and Carmichael numbers comprise those even rarer composite numbers that pass as prime numbers no matter what value of b is chosen.
In other words, Fermat's test provides a remarkably efficient means of establishing that a given number hundreds of digits long is composite. But because certain numbers slip through, it can't serve as a definitive test of primality.
"This has made things very frustrating for primality testers," Williams says. "Although we have ways now of getting around this, it turns out the nicest, simplest way of attempting to get a primality test fails utterly because of the Carmichael numbers."
How common are Carmichael numbers? The smallest is 561. There are only 43 smaller than 1 million, and a total of 105,212 among the first [10.sup.15] whole numbers.
Until the beginning of this year, Carmichael numbers seemed both scarce and hard to find. Then Alford discovered a surprisingly simple, practical method for identifying Carmichael numbers in vast quantities. "He found a way to produce huge numbers of Carmichael numbers with hardly any work," Pomerance says. "It worked out beautifully."
The ease with which it was possible to demonstrate the existence of so many Carmichael numbers spurred Alford's colleagues to return to the long-standing question of whether there was an unlimited supply of these numbers. "Certain faculty members, here at the University of Georgia, taunted the number theory group that...surely Alford's idea should provide sufficient impetus to finally prove that there are infinitely many Carmichael numbers," Granville says. "And indeed it did."
Filling in the details of the resulting proof required sophisticated techniques and special theorems drawn from several fields of mathematics. "Maybe with this theorem under our belts, it may be possible for us or other people to start finding examples...of composite numbers that would also get through [other] tests," Pomerance notes.
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|Title Annotation:||identifying Carmichael numbers|
|Date:||Sep 19, 1992|
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