Transition Year Maths 12. Fermat’s Last Theorem (and other conjectures)
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.
Fermat’s Conjecture
Fermat’s famous conjecture stated that for x3 x4 x5 xn
or or or
+ y3 + y4 + y5 + yn
= z3 = z4 = z5 = zn
where n is any number greater than 2; there are no whole number solutions
Fermat wrote in the margin of his copy of Arithmetica around 350 years ago, ‘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.’
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.
We have already looked at Pythagoras’s Theorem, x2 like 32
+ 42 = 52 or 52 + 122 = 132 + y2 = z2 . This has easy solutions
thousands of mathematicians have spent years looking at x3 looks simple, but it is very difficult to prove that it has no solution.
PROJECT 12.1 Whole Numbers & Equations
No one has ever found any whole numbers which work in the equation x3
+ y3 = z3 , but using numbers
between 1 and 10 there is one solution which is only wrong by 1. Try to find it. x, y and z are three different numbers between 1 and 10, x3
+ y3 = z3 –1
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. Only a handful of people in the world would have any chance of understanding it or verifying that it is in fact true. Very few people even with a degree in mathematics could understand the first one of the 130 pages.
114
. It was proven 2,600 years ago. In the last 350 years + y3
= z3 . This equation
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