Fermat's Really Last Theorem

Copyright ©1997 by Paul Niquette. All rights reserved.

There is only one pair of numbers
that will satisfy the equation
x = yx/y

Is this conjecture true?

French Mathematician, Pierre de Fermat, 1601 - 1665, is often called the founder of the modern theory of numbers.  He was one of the two leading mathematicians of the first half of the 17th century, the other being René Descartes.  Fermat became inspired to his researches in number theory by an edition in 1621 of the Arithmetic of Diophantus, the Greek mathematician of the 3rd century AD -- more than two and a half centuries before the first appearance of Puzzles with a Purpose.

Fermat discovered new results in the so-called higher arithmetic, many of which concerned properties of prime numbers. One of the most elegant of these had been the theorem that every prime of the form...

4n + 1
...is uniquely expressible as the sum of two squares.  A more important result, now known as Fermat's lesser theorem, asserts that if p is a prime number and if a is any positive integer, then...
ap - a
...is divisible by p. It was Gottfried Leibniz, the 17th-century German mathematician and philosopher, and Leonhard Euler, the 18th-century Swiss mathematician, who took the trouble to prove this assertion.

Fermat liked to do demonstrations of his theorems using a device that he called his method of "infinite descent," an inverted form of reasoning by recurrence or mathematical induction. Watch for publication of the series entitle The Imitation Game, in which an "infinite regress" is revealed in a famous mid-twentieth-century gedanken (ahem) formulated by Alan Turing.

One unproved conjecture by Fermat turned out to be false. In 1640, Fermat sent an e-mail message (just kidding) to mathematicians and sophisticated thinkers of the day. He put forward his belief that what became known as "Fermat Numbers" are necessarily prime.  They are of the form...

22n+ 1
A century later Euler showed that 232 + 1 = 4,294,967,297 is not a prime -- that it has 641 as a factor, the other being 6,700,417. Give that an [expletive deleted], Pierre.  But do not despair: a worldwide challenge continues into the 21st Century, finding divisors for Fermat numbers.  Meanwhile, Carl Friedrich Gauss in 1796 found an unexpected property --  that a regular polygon of N sides is constructible in a Euclidean sense (you know, with calipers and straight-edge, as they used to teach fifth graders) if N is a Fermat prime or a product of distinct Fermat primes. Tell your friends you read that first in Puzzles with a Purpose.

You will find more interesting observations about Prime Numbers at Prime Numbers are Odd.

By far the best known of Fermat's many theorems is a problem known as his "last theorem." This appeared in the margin of his copy of Diophantus' Arithmetica and states that the equation xn + yn = zn, where x, y, z, n are positive integers, has no solution if n>2.

This theorem remained unsolved until the late 20th century. Andrew Wiles devoted much of his entire career to proving Fermat's Last Theorem, regarded by some as the world's most famous mathematical problem. In 1993, he made front-page headlines when he announced a proof of the problem, but ~~ tah-dum ~~ an error was discovered in his calculation, which jeopardized his life's work. Andrew Wiles came to terms with the mistake, and eventually went on to achieve his life's ambition.  Thousands of articles on Fermat's Last Theorem can be found on the World Wide Web, including this one at the Nova site.

Fermat was the most productive mathematician of his day, but his influence was limited by his reluctance to publish.  And now, published here for the first time in history, mathematicians and sophisticated thinkers of today have this unproved conjecture, which is thought to be Fermat's Really Last Theorem. And quite possible false.