The problem posed by Diophantus was to express a square rational number ( a fraction), as the sum of 2 other square number. Today these are called Pythagorean numbers and have been known and studied in Europe, the Middle East, India and China for over a thousand years.
The problem to which Fermat believed he had found an ingenious solution was the following:
Given positive integers, x,y,z and n , where n is larger than 2, then the equation:
X**n + Y**n = Z**n , has no solutions.
This isolated problem, seemingly unrelated to any larger class of problems. has attracted the attention of thousands of persons , from the famous to the demented, or any combination of the above, for over 300 years. On June 23, 1993, Andrew J. Wiles , a mathematician at the Institute for Advanced Study in Princeton, New Jersey, announced that he had, after 7 years of work using very advanced methods of modern mathematics, proven Fermat's conjecture. Errors and gaps were found in the proof, but it appears that all of these have been patched up or accounted for, and his work was finally published in the Annals of Mathematics of May, 1995.
" I have" he states with great seriousnessness, "a proof of Fermat's Last Theorem."
For this story and the following I am indebted to Miss Capi Corrales, professor of mathematics at the University of Madrid. The Retiro Park of this same city is the meeting place of a club of elderly retirees. Every week or so they come together to renew friendships, exchange gossip, soak up the sunlight. Then they get down to the serious business that brings them together: a proof of Fermat's Last Theorem by elementary arithmetic.
An acquaintance of mine lives in a small college town on the East Coast. Even with a degree in mathematics, he is unemployed, perhaps unemployable. During a brief but traumatic participation with the U.S. Army during the Vietnamese War, he received some minor physical wounds but deep emotional scars. He now affects to hate mathematicians, despise mathematics, indeed to have contempt for all science. His view of the world is unrelieved, bitter, negative, intolerant. A great deal of his free time over the last decade or so has been devoted to finding a proof of Fermat's Last Theorem using elementary trigonometry. He accuses Andrew J. Wiles, the man who made the final effort to prove the theorem after a series of major advances in number theory starting around 1986, of cheating.
Anyone, he claims, could have proven the theorem with all that equipment at their disposal. The trick is to prove it using almost nothing. And so he toils away, claiming that the proof is just around the corner and that someday he will astonish the world.
If perchance you pay a visit to the Harvard University Mathematics Department on the 3rd floor of the Science Center, you ought to take a look at some of the things put up on the bulletin boards on the right wall of the corridor leading to the Math Library. On one of those double pronged gadgets designed to hold pages with a pair of holes punched on their edge, there is a sheaf of letters sent to Harvard faculty, filled with fantastic claims of easy proofs of problems that have resisted solution for hundreds of years, systems of astrology, mystical insights, privileged revelations, totally original systems of numbers quite unlike our integers, and so on. The crank corner, if you like, though not all of them are cranks. Some of them come from persons more like our club of retired gentlemen in Retiro Park.
A very large percentage of these letters are filled with new proofs of Fermat's Last Theorem. I went there just a few weeks ago and, because nobody seemed to want them, I took away half a dozen 'amazing proofs' of the problem that had defied the efforts of the greatest mathematicians for 356 years. I have studied these and compared them with other examples that have come to my attention over the years, and discovered that such announcements have a predictable structure :
For example, I have in front of me a letter in which the author asserts that a particular Hindu/Buddhist meditation device led him to the remarkable solution that he now presents to us. Another letter begins:
Conventional wisdom says that a negative number is a number that when added to a positive number of the same amount will TAKE AWAY that positive number. But the negative number always disappears because a positive number, ( as you already know) , is a number that when added to a negative number of the same amount will TAKE AWAY that negative number. Therefore the sum of all numbers ( including zero) that exist ought to add up to zero!!.."
In 1637, when he stated his conjecture, Fermat himself believed that he had a proof. I will discuss this in more detail in another section, but here I just want to point out that the problem is so easy to state, and appears to be so tractable, that even some of the great mathematicians have been seduced into believing that they had succeeded in whole or in part in disposing of it. For 300 years it has been the mathematical playground par excellence for every kind of beginner, amateur, crank , visionary, poet, psychotic, egomaniac, failed scientist, inventor or engineer, etc., etc.... There can be no doubt that , with the publication of the definitive and exhaustively corrected proof of Andrew J. Wiles, in the Annals of Mathematics, a glory has passed away from the world. What are those people who dream of a quick shot to scientific immortality going to work on, now that the sacred cow has been slaughtered? There are many other very famous conjectures, such as the Riemann hypothesis and the Poincare hypothesis, which would bring as much or even greater fame to the person(s) who settle them. Unfortunately, they can only be stated in a language which requires some exposure to college mathematics (2) , whearas the statement of Fermat's Last Theorem only requires a bit of high school algebra.
A permanent growth area for Alice-in-Wonderland mathematical physics still remains: the elaboration of crude or elaborate "refutations" of Einstein's Theory of Relativity. These are often propounded by good scientists who don't happen to have much knowledge of relativity theory. Alfred North Whitehead, an English academic with a good literary style and a few things to say, who has been given the title of "philosopher" by other academics, English and otherwise, wrote an intriguing book, Process and Reality, in which he propounds an alternative to Einstein's theory that is so abstruse that I've never met anyone who understood what he was trying to say.
The poet Charles Olson, for whom I do have great respect, studied Process and Reality for awhile and extracted several noteworthy insights from it. Olson didn't read the whole work: like so many other "great books of the Western canon", this is probably an impossible feat. [By the way, there is a Charles Olson celebration this Saturday, August 12, in Gloucester, Mass., the town that he loved and wrote so much about, that goes on the entire day. I unfortunately cannot attend, being duty bound to persist in my coverage of this conference, but I hope that someone will go there and write up an account of it for The Tangled Web, or Ferment, or both.]
Most "refutations" of Einstein have, alas, the same status as most "proofs" of Fermat's Last Theorem.
x**(2p) + y**(2p) = z**(2p)
has no solutions in positive integers x,y,z for any prime p that does not divide x ,y or z, what is usually called "the first case". This is almost the proof of the entire theorem, if only we could eliminate the square and generalize to the second case. Furthermore this far reaching result was proven by what one would call today very elementary methods. In fact, it takes little more than a page in the book by Paulo Ribenboim 13 Lectures On Fermat's Last Theorem, Springer-Verlag 1979.
I will discuss the exotic structure, if one may call it that, of the isolated, yet often insightful, what are sometimes called deep, partial results leading to the proof of Fermat's Last Theorem, in another section.
In response to an inquiry by Ribenboim about the current status of the prize, Dr. Schlichting replied:
Dear Sir:( The full text of the letter may be read on pages 15-16 of Ribenboim's book) ....the prize [is] now worth a little bit more than 10,000 DM. There is no count of the total number of "solutions' submitted so far. In the first year ( 1907-1908) 621 solutions were registered in the files of the Akademie, and today they have stored about 3 meters of correspondence concerning the Fermat problem.....the secretary of the Akademie divides the arriving manuscripts into (1) complete nonsense, which is sent back immediately, and into (2) material which looks like mathematics. The second part is given to the mathematics department..........at the moment I am the victim. There are about 3 to 4 letters to answer per month, and there is a lot of funny and curious material arriving, e.g., like the one sending the first half of his solution and promising the second if we would pay 1000 DM in advance; or another one, who promised me 10 per cent of his profits from publications, radio and TV interviews after he got famous, if only I would support him now; if not, he threatened to send it to a Russian magazine to deprive us of the glory of discovering him. From time to time someone appears in Göttingen and insists on personal discussion.
Nearly all "solutions" are written on a very elementary level ( using the notions of high school mathematics and perhaps some undigested papers in number theory), but can nevertheless be very complicated to understand. Socially the senders are often persons with a technical education but a failed career who try to find success with a proof of the Fermat problem. I gave some of the manuscripts to physicians who diagnosed heavy schizophrenia.....
Back to Fermat's Last Theorem: Conference Report