Fermat’s Last Theorem

Today is world book day, so I am compiling the story of Fermat’s last theorem as it’s described in the book Fermat’s Last Theorem written by Simon Singh. I loved the way he has written the book. With every single chapter you come closer and closer to the story of final proof.

According to Fermat’s last theorem

x^n + y^n = z^n does not have any whole number solutions for all n>2

So according to the theorem this equation does have solutions but not whole number solutions. Whole numbers are a set of numbers starting from 0 to infinity, also called positive integers.

When we take n=2 Fermat’s equation turns into another famous equation which is related to Pythagoras theorem. The solution to this equation is called Pythagorean triples and there are infinitely many Pythagorean triples or infinitely many whole number solutions to the equation

x^2 + y^2 = z^2

But according to Fermat’s Theorem there are no solutions to similar equations with power greater than 2.

So the story starts when Fermat left a marginal note in a book he was reading that x^n + y^n = z^n has no whole number solutions but the margin is too small to outline the whole proof.

  1. Fermat didn’t only leave the marginal note saying he had the proof but he actually proved the theorem for n=4 later in order to solve another problem. He used a form of proof by contradiction called “method of infinite descent”.
  2. Euler used imaginary numbers and the infinite descent method to prove Fermat’s theorem for n=3 but it was incomplete and was later completed.
  3. FLT was proved for n=3 and n=4 so it can now be proved for other cases where n is a multiple of 3 or 4. Another more important conclusion was, since 3 is a prime number and according to fundamental theorem of arithmetic every number is a multiple of prime numbers, if one could only prove that the FLT is true for prime numbers it will be proved for other numbers just because these remaining numbers are just multiple of primes. But prime numbers are also infinite just like the whole numbers so it doesn’t make any difference.
  4. 75 years later Sophie Germain gave a method to prove FLT for all prime numbers p such that 2p+1 is also a prime. So the condition is, if FLT is true for p = 3 then it will be true for 2p + 1 = 7 which is also a prime. Dirichlet and Legendre, using her method proved FLT for n=5 and fourteen years later Gabriel Lame proved it for n=7.
  5. Taniyama and Shimura showed that elliptic curves and modular forms were one of the same things.
  6. In 1984 Gerhard Frey turned Fermat’s equation into an elliptic curve which is called Frey’s curve. He proposed a series of steps that one could follow in order to prove Fermat’s last theorem.

If Taniyama-Shimura conjecture is true then every elliptic curve must be related to a modular form. So Frey’s curve won’t exist because there won’t be a corresponding modular form. If Frey’s curve won’t exist then this will imply that there are no solutions either. So FLT will be true.

  1. Ken Ribet proved that Frey’s elliptic curve is not modular.
  2. So now one has to prove the Taniyama-Shimura conjecture in order to prove FLT. And it was done by Andrew Wiles in 1993 after years of patience and hard work. But there was an error in the proof which was later resolved and the final proof was published in 1995.

 

Now if we consider a similar equation when n=1

a + b = c

where a, b and c are relatively prime that is, divisor of one number cannot be the divisor of another. So divisors of each of these numbers must be different.

If this condition is satisfied then a conjecture named abc conjecture says that

Radical of (abc)^k > c

Radical of the number abc (after multiplying three numbers together) is the multiplication of different prime factors of each of these numbers.

There are exceptions to this conjecture. If k=1, there are infinitely may exceptions and if k>1, there are finitely many exceptions.

Surprisingly this equation has a beautiful story just like Fermat’s Last Theorem. And whenever its proof will be found it will also be a historic moment.

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