quit pierre de fermat fermat’s last conjecture prime numbers euler’s conjecture
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Pierre de Fermat
Fermat’s Last Conjecture
Prime Numbers
Euler’s Conjecture
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Pierre de FermatPierre 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.
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• Fermat’s famous conjecture stated that for
x3 + y3 = z3
or x4 + y4 = z4
or x5 + y5 = z5
or xn + yn = zn
there are no whole number solutions.
Fermat’s Last ConjectureFermat’s Last Conjecture
where n > 2
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• 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 ConjectureFermat’s Last Conjecture
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• 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 €1,000,000 today) to anyone who could prove Fermat’s last theorem before 2007.
Fermat’s Last ConjectureFermat’s Last Conjecture
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• We have already looked at Pythagoras’s Theorem x2 + y2 = z2.
• 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 x3 + y3 = z3. • This equation looks simple, but it is very difficult
to prove that it has no solution.
Fermat’s Last ConjectureFermat’s Last Conjecture
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• 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.
Fermat’s Last ConjectureFermat’s Last Conjecture
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• 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.
Prime NumbersPrime Numbers
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1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
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• There was another conjecture called Euler’s Conjecture which stated that there were no solutions to the equation:
x4 + y4 + z4 = w4
• Then amazingly a man called Naom Elkies in 1988 found
Euler’s ConjectureEuler’s Conjecture
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• 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.
Goldbach’s ConjectureGoldbach’s Conjecture
Yes No
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