Quit Pierre de Fermats Last Conjecture Prime Numbers

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Pierre de Fermat’s Last Conjecture Prime Numbers Euler’s Conjecture Quit

Pierre de Fermat • Pierre de Fermat was born in France in 1601 and was probably the world’s best amateur mathematician. • He invented and proved many theorems in mathematics. A statement in maths is called a conjecture until it is proven. Quit

Fermat’s Last Conjecture • Fermat’s famous conjecture stated that for 3 3 3 x +y =z or x 4 + y 4 = z 4 5 5 5 or x + y = z where n > 2 or xn + yn = zn there are no whole number solutions. Quit

Fermat’s Last Conjecture Quit • Fermat wrote in the margin of his copy of Arithmetica in 1670, ‘To resolve a cube into the sum of two cubes, or a fourth power to the sum of two fourth powers, or in general any power higher than the second into two of the same kind, is impossible, of which fact I have found a remarkable proof. The margin is too small to contain it. ’

Fermat’s Last Conjecture • All of Fermat’s other theorems have been verified years ago, but this one has proved very difficult. • In 1908 Dr Paul Wolfskehl offered a prize of 100, 000 Deutschmarks (worth about € 1, 000 today) to anyone who could prove Fermat’s last theorem before 2007. Quit

Fermat’s Last Conjecture • • • Quit We have already looked at Pythagoras’s Theorem x 2 + y 2 = z 2. This has easy solutions like 32 + 42 = 52 or 52 + 122 = 132. It was proven 2, 600 years ago. In the last 350 years thousands of mathematicians have spent years looking at x 3 + y 3 = z 3. This equation looks simple, but it is very difficult to prove that it has no solution.

Fermat’s Last Conjecture • • • Quit Most people believe that if Fermat had a proof, it probably had an error in it. A man called Andrew Wiles finally proved the conjecture in 1994, after working on it for almost all of his adult life, often in secret. The proof takes up 130 pages of small print.

Prime Numbers • • • Quit A prime number is a natural number which has no proper factors. There is a lot of interest in prime numbers, especially to cryptographers. Codes based on multiplying large prime numbers together are very difficult to break. To some extent the success of computer-based financial transactions depends on the security provided by such codes.

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Euler’s Conjecture • • Quit There was another conjecture called Euler’s Conjecture which stated that there were no solutions to the equation: x 4 + y 4 + z 4 = w 4 Then amazingly a man called Naom Elkies in 1988 found

Goldbach’s Conjecture • • Quit Goldbach’s Conjecture was proposed by Christian Goldbach (1690 – 1764). When Andrew Wiles was in Dublin in 2003 he was asked if he was now working on Goldbach’s Conjecture – he did not reply. The conjecture states, ‘Every even number greater than 2 is the sum of two prime numbers’. Pick any even number. 20 can be written as the sum of 13 and 7; two prime numbers.

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