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3. c2c2 a2a2 b2b2 a 2 + b 2 = c 2 Fermat’s Last Theorem a b c In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Pythagoras of Samos (6C BC) Pierre de Fermat (1601 – 1675) Fermat’s Last Theorem is strongly linked to Pythagoras’ Theorem. In particular, solutions to the equation a 2 + b 2 = c 2 where a, b and c are all positive integers.

4. 25 9 16 3 2 + 4 2 = 5 2 9 + 16 = 25 A Pythagorean Triple 3 4 5 3, 4, 5 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

5. 169 144 25 5 2 + 12 2 = 13 2 25 + 144 = 169 A 2nd Pythagorean Triple 5, 12, 13 5 12 13 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

6. 625 576 49 7 2 + 24 2 = 25 2 49 + 576 = 625 7 24 25 A 3 rd Pythagorean Triple 7, 24, 25 In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

7. 22122021 10 18118019 9 14514417 8 11311215 7 858413 6 616011 5 41409 4 25247 3 13125 2 543 1 ??2n+1n  There are an infinite number of triples of this type.  Pythagorean Triples (Shortest side odd) 2n 2 + 2n 2n 2 + 2n + 1

8. 48548344 10 40139940 9 32532336 8 25725532 7 19719528 6 14514324 5 1019920 4 656316 3 373512 2 17158 1 ??4n+4n  There are an infinite number of triples of this type.  Pythagorean Triples (Shortest side even) 4n 2 + 8n + 3 4n 2 + 8n + 5

9. There are an infinite number of solutions to a 2 + b 2 = c 2 3232 9 + 4242 16 = 5252 25 6363 216 + = 8383 512 9 3 - 1 728 Whilst reading page 6 of a copy of Diophantus’s “Arithmetica”, Fermat became intrigued by the sheer variety of Pythagorean triples discussed in his book. He began to think about solutions to a n + b n = c n for n > 2 Diophantus was a Greek mathematician who lived from about 200 to 280 AD Fermat noted in the margin of the book that there were no solutions above n = 2 and that he had a truly marvellous proof of it. What about a 3 + b 3 = c 3 or a 4 + b 4 = c 4 ?

10. Read the full story of how the English mathematician, Andrew Wiles, eventually solved Fermat’s Last Theorem, in Simon Singh’s superb book. He takes you on an amazing, easy to read odyssey of mathematical history, from Ancient Greece to the present day. The Story of a Riddle that Confounded the World’s Greatest Minds for 358 Years.

11. Fermat’s Last Theorem Over three hundred and fifty years were to pass before a mild- mannered Englishman finally cracked the mystery in 1995. Fermat by then was far more than a theorem. Whole lives had been devoted to the quest for a solution. There was Sophie Germain who had to take on the identity of a man to conduct research into the field forbidden to females. The dashing Everiste Galois, scribbled down the results of his research deep into the night before sauntering out to die in a duel. The Japanese genius Yutaka Taniyama, killed himself in despair, while the German industrialist Paul Wolfskehl claimed Fermat had saved him from suicide. “I HAVE DISCOVERED A TRULY MARVELOUS PROOF, WHICH THIS MARGIN IS TOO NARROW TO CONTAIN..” With these tantalising words the seventeenth-century French mathematician, Pierre de Fermat threw down the gauntlet to future generations. Fermat’s last theorem looked simple enough for a child to solve, yet the finest mathematical minds would be baffled by the search for the proof. Andrew Wiles had dreamed of proving Fermat ever since he first read about the theorem as a boy of ten in his local library. Whilst the hopes of others had been dashed, his dream was destined to come true – but only after years of toil and frustration, of exhilarating breakthrough and crashing disappointment. The true story of how mathematics’ most challenging problem was made to yield up its secrets is a thrilling tale of endurance, ingenuity and inspiration.