Prove a Theorem, Win $1,000,000!

ByJohn Allen Paulos

April 1, 2000 -- — One generally doesn’t speak the words “prime numbers” and “seven-figure prizes” in the same breath.

But don’t tell that to the publishers of Uncle Petros and Goldbach’sConjecture, an engaging first novel by Greek author ApostolosDoxiadis.

Before getting to the money, here’s a quick synopsis of the story: The narrator tells of his Uncle Petros, whom he initially thinks of as the eccentric black sheep of the family.

Slowly, Uncle Petros is revealed to be a character of complexity and nuance, havingdevoted his considerable mathematical talents and much of his life to afutile effort to prove a classic unsolved problem. His solitary effortsgive one a taste of the delight and the despair of mathematical research.

Goldbach’s Conjecture, Uncle Petros’ holy grail, is startlinglysimple to state:

Any even number greater than 2 is the sum of two prime numbers.

Remember that a prime number is a positive whole number that is divisible only by two numbers: itselfand 1; thus 5 is a prime, but 6, which is divisible by 2 and 3, is not. The number 1 is not considered prime.

Check out the claim. Pick an even number at random and try to find twoprimes which add up to it. Certainly, 6 = 3 + 3, 20 = 13 + 7, and 97 + 23 =120.(This, of course, is not a proof.) The conjecture that this works for everyeven number greater than 2 was proposed in 1742 by Prussian mathematicianChristian Goldbach. To this day it remains unproved despite the efforts ofsome of the world’s best mathematicians.

Number theory, the branch of mathematics that studies primenumbers and other ethereal aspects of theintegers (whole numbers), contains many problems that are easy to state andyet resistant, so far, to the efforts of all.

The Twin Primes Conjecture isanother:

There are an infinite number of prime pairs, prime numbers that differby 2.

Examples are 5 and 7, 11 and 13, 17 and 19, 29 and 31, and,presumably, infinitely many more.

Of more contemporary origin is the so-called Collatz Conjecture,sometimes called the 3x + 1 problem.

Choose any whole number. (Take 13, for example) If it is odd,multiply it by 3 and add 1. (3 times 13 plus 1 equals 40.) If it is even, divide it by 2. (40 divided by 2 is 20.) Continue thisprocedure with each resulting number and the conjecture is that the sequencethus generated always ends up 4, 2, 1, 4, 2, 1, 4, ... The sequence starting with 13, produces 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ....

Every number that’s been tried (up to about 27 quadrillion) ultimatelycycles back to 4, 2, 1, but there is still no proof that every number does.

Like the recently proved Fermat’s Last Theorem (xn + yn = zn, where x, y, z, n are all integers, has no solutions for n > 2), these conjecturesare tantalizing and can sometimes become, if one is not careful,all-consuming obsessions. Such an obsession is the fate of Uncle Petros. His brothers thinklittle of him and of his quixotic attempt to prove Goldbach’s Conjecture.

Petros in turn has disdain for their petty concern with the family businessin Athens. Interestingly, Petros also has a low regard for appliedmathematics, which he compares to glorified “grocery bill” calculations andwhich, he believes, shares none of the austere beauty of pure number theory.

This mutual contempt between mathematicians and more practical sorts has along history. The British mathematician G.H. Hardy, a colleague of the fictional Uncle Petros in the book, exulted in theuselessness of mathematics, particularly number theory.

Happily, this adversarial attitude has softened in recent years,and even number theory, arguably the most impractical area of math, hasfound important applications. Cryptographic codes, which enable the transferof trillions of dollars between banks, businesses, and governments, dependcritically on number theory.

They depend, in particular, on the simple factthat multiplying two large prime numbers together is easy, but factoring alarge number (say one having 100 digits) into prime factors isextraordinarily difficult and time-consuming.

Finally, I come to the million dollar contest. The U.S. publisher ofUncle Petros and Goldbach’s Conjecture has promised $1 millionto the first person to prove the conjecture, provided the proofappears in a reputable mathematics journal before 2004.

The late, great number theorist Paul Erdos used to offer small monetary prizes to anyonesolving this or that problem, but he didn’t have to pay up often. If I werethe publisher, I wouldn’t worry about the offer’s financial risk, but Iwould be apprehensive about the torrent of false proofs that will soon be headingtheir way.

Professor of mathematics at Temple University, John AllenPaulos is the author of several books, including AMathematician Reads the Newspaper and, most recently, I Think, Therefore I Laugh. His “Who’s Counting?” column on ABCNEWS.com appears on the first day of every month.

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