Here is a compilation of the assertions made by Fermat about numbers. Almost all of them come from the texts you read last week, but I couldn't resist adding a couple.

Some natural questions to ask about each of these assertions are:

- Where did Fermat find this?
- Is it true?
- Can I prove it's true? (or false?)
- Why is this an interesting question?

As you can see from Pascal's letter, not everyone thought these *were*
interesting questions!

- The number is always a prime number.
- Every number is the sum of at most three triangular numbers.
*Note: if we declare that 0 is a triangular (square, pentagonal, ...) then we can drop the ``at most'' from all these assertions. People in Fermat's time were still a little suspicious of 0 as a number.* - Every number is the sum of at most four squares.
- Every number is the sum of at most five pentagonal numbers.
- Etc.
- A prime number of the form
*p*=4*k*+1 (one more than a multiple of four) can be obtained as the sum of two squares, i.e., , where*u*and*v*are whole numbers. - A prime number of the form
*p*=3*k*+1 can be expressed as , where*u*and*v*are whole numbers. - A prime number which is either of the form
*p*=8*k*+1 or*p*=8*k*+3 can be expressed as , where*u*and*v*are whole numbers. - A right triangle whose sides have whole number lengths cannot have
area equal to a square. (For example, if the triangle has sides
equal to 3, 4, and 5 units then its area is 6, which isn't a square.
- Fix a prime number
*p*, and consider the sequence of powers of 2:Then:

- There is always a smallest exponent
*n*such that is a multiple of*p*. - This
*n*is always a divisor of*p*-1. - If
*m*is any other exponent such that is a multiple of*p*, then*m*is a multiple of*n*.

- There is always a smallest exponent
- The statements above remain true when we replace 2 with any other
number that is not divisible by our prime number
*p*. - Given any number
*N*that is not a square, there exist infinitely many pairs of whole numbers*x*and*y*such that . - In the previous problem, there exists an explicit ``mechanical''
procedure for finding all the solutions
*x*and*y*. - No cube can be written as a sum of two cubes.
- No fourth power can be written as a sum of two fourth powers.
- In general, if
*n*is bigger than 2, no*n*-th power can be written as the sum of two*n*-th powers.

Fri Nov 21 13:32:48 EST 1997