When science meets a deadlock, money should talk

By Lim Tri Santosa

BANDUNG (JP): The resolution of apparently pure mathematical problems always affects non-mathematicians eventually. Without maths, there would be no computers, no semiconductors, no global positioning systems, no space race, no gene studies, no telecommunications.

Like the elements in chemistry, prime numbers serve as building blocks in the mathematics of whole numbers. Evenly divisible only by themselves and one, primes are a rich source of speculative ideas that mathematicians often find simple to state but difficult to prove.

Prime numbers provide a rich source of speculative mathematical ideas. Today, prime numbers are fascinating but they are also of commercial importance, since the best commercial and military ciphers depend on their properties. Number theory, the branch of mathematics that studies prime numbers and other ethereal aspects of the integers (whole numbers), contains many problems that are easy to state and yet resistant, so far, to the efforts of all.

In 1742, a little-known mathematician named Christian Goldbach, professor of mathematics and historian of the Imperial Academy at St. Petersburg and who also was a math tutor to Tsar Peter II, speculated that every even number greater than four could be expressed as the sum of two prime numbers.

Goldbach's conjecture is easy to verify for small numbers: 6=3+3, 8=3+5, 10=3+7=5+5, 24=5+19=7+17=11+13, and so on. This, of course, is not proof of an infinite given.

The conjecture is looking safe so far. Not only is each even number the sum of two primes, but the number of pairs of primes tends to increase. This trend seems to continue. But no one has ever proved that this goes on forever. No reasonable mathematician seriously doubts the validity of Goldbach's conjecture.

Notice that there can be more than one Goldbach pair (10 has 2 pairs, 24 has 3 pairs). The conjecture says only that there is at least one, and has nothing to say about whether there may be more. You can explore the Goldbach conjecture yourself with the Goldbach calculator at http://pass.maths.org/issue2/xfile/ for 4 < n < 10,000. Simply enter an even integer, n, greater than 4 and the calculator will find all the Goldbach pairs.

But mathematical proof is not a matter of what seems reasonable but rather a product of pure reason. Thus far Goldbach's little observation has resisted the reasoning power of the greatest minds in mathematics. To this day it remains unproved despite the efforts of some of the world's best mathematicians. It's incredibly difficult to prove that Goldbach's guess holds true for all of the infinite host of even numbers.

In 1973, Chinese mathematician J. R. Chen (source: britanicca.com) showed that every sufficiently large even number is either the sum of two primes or of a prime and a near prime (take 24=3+21, 3 is a prime, and 21=3x7). A near prime is a number that is the product of two primes, like 91=7x13 or 4=2x2. No one knows just how large, "sufficiently large", actually is. But close enough isn't good enough in mathematics, so the original conjecture remains unproven.

In 1998, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified the Goldbach's conjecture up to 400,000,000,000,000 (400 trillion) by brute force with the help of a Cray C90 super-computer and a number of workstations.

The smaller prime needed in the considered interval is never larger than 5,569 and is needed for the partition 389,965,026,819,938 = 5,569 + 389,965,026,814,369. (source: www.andrews.edu/~calkins/math/biograph/biogoldb.htm)

For mathematicians, a proof is not a guess-work evidence (read: a conjecture), based on calculating numbers by brute force and exploiting a computer system up to a certain highest finite number. A math proof is an absolute pattern without any exception beyond the infinity.

On March 15, 2000, the British and American publishers offer of a prize for solving Goldbach's conjecture is both a clever publicity stunt and also in the finest traditions of mathematics. They will pay US$1 million to anyone who can prove Goldbach's conjecture before March 15, 2002. More precisely, before offering their prize, Faber & Faber and Bloomsbury Books, the British and American publishers of Uncle Petros and Goldbach's Conjecture, secured insurance with Lloyd's. The rules of the contest can be found at www.faber.co.uk.

Mathematicians once believed that the truth or falsity of any mathematical proposition could be decided by applying the rules of mathematical logic.

For them, the techniques invented and principles illuminated by pursuing the proof of a hard problem are the real rewards. Mathematical proofs of conjectures, however, require more than overwhelming numerical evidence.

A prize of $1 million could well stimulate new efforts to crack this famous problem. Thus if you can't do this for the love of science, do it for the love of money. Fame is nice too. Good luck.