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Fermat’s Last Theorem Dr. S. Lawrence ©2005

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**Who do we start from? Pierre de Fermat 1601-1665**

civil servant all his life mathematics his passion and a pastime

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**Or do we go even further back?**

Fermat’s theorem is intriguing and exciting because it is based on one of the simplest and most beautiful theorems which almost every one knows about: Pythagoras’ theorem. Pythagoras theorem states that …

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**Pythagoras’ Theorem x 2 + y 2 = z 2**

Or, in words: the square on hypotenuse is equal to the sum of squares on two other sides.

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Fermat’s Theorem Fermat conjectured however, that this, if applied to cube or any other power would not work, in other words x n + y n = z n has no nontrivial whole number solutions for n > 2. You can try for yourself on few example.

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**Why is it a theorem and not a conjecture?**

Had Fermat been satisfied to just write it down as we did on the previous page, it would have remained a conjecture. However, he wrote, in a margin of a book he was studying at the time (famous Arithmetica of Diophantus) that he just did not have enough space to write the simple proof, although he was in a possession of one.

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**The exact text Fermat wrote exactly this (translated from Latin):**

“It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.”

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“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.”

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Fermat’s Last Theorem What this actually mean in algebraic terms is (what we said on the web page which led you here) that although Pythagoras’ Theorem is quite valid, the equivalent for any other power larger than 2 x n + y n = z n for n = 3, 4, 5,.. makes the equation invalid.

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Hilbert’s Challenge In 1900 Hilbert, at the International Congress of Mathematicians, challenged mathematicians around the world to solve Fermat’s Last Theorem.

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Some solutions Few confirmations of Fermat’s Theorem were already known by the 1900; the quest to prove other cases continued all the way up to 1994 when Andrew Wiles proved Fermat’s Last Theorem by the Method of Exhaustion (meaning that he tried with all the cases that the computer could ‘think of’ and none have proved Fermat’s Last Theorem to be wrong).

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**Nothing left to do? Not so…**

But you can still work on it… Think of a nicer or more ‘elegant’ proof to confirm (or disprove!) Fermat’s Last Theorem If you do, you can guarantee yourself to have a place at the most prestigious department of mathematics at the most prestigious university in the world! Good luck!!!

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SECTION 5-3 Selected Topics from Number Theory Slide 5-3-1.

SECTION 5-3 Selected Topics from Number Theory Slide 5-3-1.

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