Zeroing in on an infinite number of primes.
More than 2,000 years ago, Euclid of Alexandria proved that there is an infinite number of primes among whole numbers. Now, two mathematicians have shown that the supply of primes is also unlimited in a particular subset of the whole numbers. Along the way, they developed new, potentially powerful mathematical techniques for probing some of the mysteries of primes, including their rather haphazard distribution.
John Friedlander of the University of Toronto in Scarborough, Ontario and Henryk Iwaniec of Rutgers University in New Brunswick, N.J., report their findings in a pair of papers scheduled for publication in the ANNALS OF MATHEMATICS.
"It's a tremendous achievement," says Peter C. Sarnak of Princeton University.
Andrew J. Granville of the University of Georgia in Athens adds, "They've succeeded in solving a problem that we've been stuck on for about 100 years."
The sequence of primes goes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,37, 41, and so on. Broadly speaking, primes become more scarce as numbers get larger. Nonetheless, the sequence of primes never ends, even though the gaps between successive primes, on average, get longer.
In probing the distribution of primes, mathematicians have investigated whether the number of primes among certain subsets of the whole numbers is also infinite. For example, about a century ago, they proved that there are infinitely many primes among numbers of the form [a.sup.2] + [kb.sup.2], where a and b are integers and k is a certain constant.
It's also easy to come up with simple examples of subsets in which the number of primes is finite. For instance, among even numbers, only 2 is prime.
"The general belief is that, whenever you write down some definition of a sequence, you will have infinitely many primes unless there's some obvious reason why you don't," Sarnak says. "But there are very few cases in which [the infinitude of primes] can be proved."
Friedlander and Iwaniec tackled the question of whether there are infinitely many primes among numbers of the form [a.sup.2] + [b.sup.4]. These numbers are relatively scarce among whole numbers in general. Mathematicians describe such a sequence as sparse.
Thus, among whole numbers up to 100, there are 18 numbers of the required form: 2, 5, 10, 17, 20, 25, 26, 32, 37, 41, 50, 52, 65, 80, 82, 85, 90, and 97, of which only six--2, 5, 17, 37, 41, and 97--are primes. Among whole numbers up to 1 trillion, there are fewer than 1 billion numbers of the form [a.sup.2] + [b.sup.4].
Despite the relative scarcity of these numbers, Friedlander and Iwaniec not only proved that this particular sparse subset includes an infinite number of primes but also accurately determined the frequency of primes in the sequence--that is, the likely number of primes within a given range.
"Everybody expected that result," Sarnak says. "The surprise was that they could prove it. Even for experts, it seemed way out of reach."
The techniques developed by Friedlander and Iwaniec may turn out to be useful for tackling several other key questions in number theory. No one has yet proved that there is an infinite number of primes among numbers of the form [a.sup.2] + 1, which would be a subset of the subset studied by Friedlander and Iwaniec.
Mathematicians would also like to prove that the number of twin primes--pairs of primes, such as 17 and 19, that differ by 2--is infinite. "That's a major unsolved problem," Sarnak comments. "It's considered to be very, very difficult."
"The question is whether the work of Friedlander and Iwaniec represents a special case or is going to open the door to lots of other things," Granville says. "It's possible it could go a long way, but it's hard to tell what's coming."
Friedlander and Iwaniec are interested in extending their approach to primes of the type a2 + b6, which would have intriguing links to mathematical equations called elliptic curves. Such curves played an important role in the recent proof of Fermat's last theorem (SN: 11/15/97, p. 310).
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|Title Annotation:||number theory|
|Date:||Jan 3, 1998|
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