The ancient mechanism of the stargate had rendered im good services,
but he wouldn't need them anymore. The flames of the inferno did no harm
to the child. Still the quarder shaped appearance was floating in front
of him; hidden inside it had undiscovered mysteries of space and time.
But some of them the child already understood and thought to master them.
How obviously - how necessary! - was the mathematical relation of the sides
of the monolith - the square sequence of 1 : 4 : 9! And how naiv it was
to assume that this series would end up only within the three dimensions!
(Arthur C. Clarke, "2001 - Odyssee im Weltraum", 1969, Heyne 1978, retranslated)


Pierre de Fermat

The Mysteries of the Powers of Integers

Fermat Portrait by Francois de Poilly

Pierre de Fermat (1601-1665), Mathematician

Pierre de Fermat was born at the 17th of August in 1601 in Beaumont de Lomagne, France. This birthday is not completely sure, but it is based on the fact that the christening happend at August 20. - After school he studied jurisprudence, and with an age of 30(33?) he became councillor at the court of Toulouse. According to mathematics, Fermat was amateur and probably self-tought. His sources were Greek texts about mathematics, most of all the book "Arithmetica" of Diophantos of Alexandria, covering problems of mathematics of the ancient times.

Despite of his amateur state Fermat - besides of Descartes (1596-1650) - has the reputation as one of the greatest mathematician of his Century, and with Descartes he is one of the developer of the geometry of axes, and with this a founder of analytical geometry. He was one of the pioneers of infinitesimal calculation, because he was working with own methods on the integration of powers with integer and fractial exponents. With this he solved tangent problems covering the integration and differentiation of curves, the finding of maxima and zero points. He had correspondence with some famous contemporaries, besides other with Blaise Pascal and Christiaan Huygens.

According to astronomy Fermat formulated the fact that a ray of light at a redirection through a mirror or a prism will take the way requiring the shortest time. This physical law is also called the "Fermat principle" and explains the refraction of light in media.

Also named after the French mathematician is the theorem, that ap-1-1 is always divisible by the prime number p without remainder, when a is an integer, which is not by p divisible. For example: for p=7 and a=6 this is 66-1=46655 and 46655/7=6665, for p=7 and a=8 the result is 86-1=262143 and 262143/7=37449, but for p=7 and a=7 the result is 76-1=117648 and 117648/7=16806rest6. The general form of this strange behaviour of the numbers is called "The Smaller Fermat Theorem".

But it was another problem of the analytical geometry, which made Fermat famous. According to the clause of Pythagoras there is a2 + b2 = c2. For this equation there are numerous integer solutions, for instance 32 + 42 = 52 or 72 + 242 = 252 or 102 + 242 = 262. But if taken three or more dimensional figures instead of two dimensional squares, an integer solution is hard to find. The great assumption of Fermat - also call "The Greater Fermat Theorem" - is


   an + bn <> cn for n>2, a,b,c>0 and n,a,b,c integers.   

This means: There is no integer solution without zero for the equation an + bn = cn if n is an integer and larger than 2. Fermat stated that he had a miracle proof for this, but he never wrote it down. Maybe it slipped him or he found an error within, because he formulated a more special form of the theorem as "Cubum autem in duos cubos" - "A cube is not in two cubes" or c3 <> a3 + b3. But maybe he thought about presenting this special form of the theorem to his contemporaries and mathematicians as a riddle as he had done it previously with other problems solved by him. He died at January the 12th, 1665 in Caustres near Toulouse. Some of his letters and side notes in books were published by his son Clement Samuel de Fermat, where also the Greater Fermat Theorem became known.

Many mathematicians tried to prove or disprove the general form of the last theorem of Fermat, besides others Gauß (1777-1855) and Euler (1707-1783). Euler was able to reject another assumption of Fermat, which was stating, that all numbers from y = 2m+1 would be prime numbers when m=2k. Euler's counter example was k=5 m=32, where y=232+1 = 4,294,367,297. But 4,294,367,297 is also the product of 6,700,417 and 641, and with this its no prime number.

Nearly threehundred and fifty years had to pass, until Fermat's great theorem got proven. In 1994 the professor at Cambridge University in England, Dr. Andrew Wiles was able to prove - after seven years of preparations and and with several other new works - that within the theoretical geometry elliptical curves are also modular formulas. This simple sounding equality implies Fermat's inequation, and with an previously found reversibility of the implication the proof for the relation of elliptical curves being modular formulas, it is also a proof of an+bn<>cn for n>2 and n,a,b,c integer.

For details about the proof of the Fermat assumption see also the script to Fermat's Last Theorem, Horizon, BBC. Another source of the text above is: Krafft/Meyer-Abich (Eds.), "Große Naturwissenschaftler - Biographisches Lexikon", Fischer 1970.


Page history:
created 1999-11-24 as himmel.01.08.html
changed 2000-01-27 translated
changed 2000-02-04 to Fermat-e.html
changed 2011-08-20 image added and link to USM removed

To the Sky over Berlin...To the astronomy links...-- jd --