ries. Analytic number theory was among his specialties, and his publications carried titles such as “Highly composite numbers” and “Proof that almost all numbers n are composed of about log(log n) prime factors.”
A legendary figure in 20th-century number theory was Paul Erdős (1913–96), a Hungarian genius known for his deep insights, his vast circle of collaborators, and his personal eccentricities. At age 18, Erdős published a much-simplified proof of a theorem of Chebyshev stating that, if n ≥ 2, then there must be a prime between n and 2n. This was the first in a string of number theoretic results that would span most of the century. In the process, Erdős—who also worked in combinatorics, graph theory, and dimension theory—published over 1,500 papers with more than 500 collaborators from around the world. He achieved this astonishing output while living more or less out of a suitcase, traveling constantly from one university to another in pursuit of new mathematics. It was not uncommon for him to arrive, unannounced, with the declaration that “My brain is open” and then to plunge into the latest problem with gusto.
Two later developments deserve mention. One was the invention of the electronic computer, whose speed has been advantageously applied to number theoretic questions. As an example, Euler once speculated that at least four fourth powers must be added together for the sum to be a fourth power. But in 1988, using a combination of mathematical insight and computer muscle, the American Noam Elkies discovered that 2,682,4404 + 15,365,6394 + 18,796,7604 = 20,615,6734—a stupendous counterexample that destroyed Euler’s conjecture. (The number on the right contains 30 digits, so there is little wonder that Euler missed it.)
Second, number theory acquired an applied flavour, for it became instrumental in designing encryption schemes widely used in government and business. These rely upon the factorization of gigantic numbers into primes—a factorization that the code’s user knows and the potential code-breaker does not. This application runs counter to the long-held perception of number theory as beautiful but essentially useless. (Seecryptology: Cryptography.)
Twentieth-century number theory reached a much-publicized climax in 1995, when Fermat’s last theorem was proved by the Englishman Andrew Wiles, with timely assistance from his British colleague Richard Taylor. Wiles succeeded where so many had failed with a 130-page proof of incredible complexity, one that certainly would not fit into any margin.
This triumph notwithstanding, number theory remains the source of many unsolved problems, some of the most perplexing of which sound innocent enough. For example:
Do any odd perfect numbers exist?
Are there infinitely many primes of the form n2 + 1 (i.e., one more than a perfect square)?
Are there infinitely many pairs of twin primes (i.e., primes that differ by 2, like 5 and 7 or 41 and 43)?
Is Goldbach’s conjecture true? (Euler failed to prove it; so has everyone since.)
Although there has been no lack of effort, these questions remain open. Perhaps, like Fermat’s last theorem, they will eventually be resolved. Or perhaps they will remain as challenges into the indefinite future. In order to spur research efforts across a wide range of mathematical disciplines, the privately funded Clay Mathematics Institute of Cambridge, Massachusetts, named seven “Millennium Prize Problems” in 2000, each with a million-dollar award for a correct solution. In any case, these mysteries justify Eric Temple Bell’s characterization of number theory as “the last great uncivilized continent of mathematics.”
The theory of numbers, then, is a vast and challenging subject as old as mathematics and as fresh as today’s news. Its problems retain their fascination because of an apparent (often deceptive) simplicity and an irresistible beauty. With such a rich and colorful history, number theory surely deserves to be called, in the famous words of Gauss, “the queen of mathematics.”