February 21

Plan: (Not followed, so copied to the next class)

  • Questions
  • Work toward Euclid's characterization of Pythagorean triples

The discussion page can be found here

Today's class was an introductory discussion of Number Theory, which mainly concerns quadratic forms.

Mersenne primes and perfect numbers (from the Greeks to GIMPS).

When is a prime the sum of two squares?

1. There are two kinds of even numbers, singly even (divisible by 2) and doubly even (divisible by 4).
2. A square of an even number is doubly even.
3. When you divide an even number by 4 you get a remainder of either 0 or 2.
4. When you divide an odd number by 4 you get either 1 or 3.

$x^2 + y^2 = p$ where $p$ is an odd prime number, conditions under which this is true:

  • Either x or y is odd and either x or y is even, they cannot both be odd or even
  • When you divide the prime by 4 the remainder is 1. Written 1(mod 4). (Gauss introduced modular arithmetic)

Proof: Either x or y is odd and either y or x is even, otherwise p would not be prime because an odd number squared is an odd number and an even number squared is an even number and the addition of an odd and an even number is even.

If x is even and y is odd by statements 1-4 the remainder when you divide $p$ by 4 is 0 + 1 and so can't be 3.

Therefore the only odd numbers that can possibly be the sum of two squares are those congruent to 1 mod 4.

Fermat proved the converse for primes: those of the form $4n+1$ can be written as the sum of two squares.

Brief history of the Fermat conjecture:

  • Fermat claimed a proof in his famous marginal comment in his copy of Bachet's edition of the work of Diophantos.
  • Euler provided a proof for n=4 using methods available to Fermat (we will read that in our text)
  • Legendre, Lame Germain made some progress.
  • Kummer showed how the failure of unique factorization in some situations destroyed proofs.
  • Frey showed how the failure of Fermat's conjecture led to a weird cubic equation (elliptic curve).
  • Taniyama's conjecture: there are no weird cubic equations.
  • Andrew Wiles proved Taniyama's conjecture and thereby proved Fermat's Last Theorem .
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