MA177--What Fermat says

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!

1. The number is always a prime number.
2. 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.

3. Every number is the sum of at most four squares.

4. Every number is the sum of at most five pentagonal numbers.

5. Etc.

6. A prime number of the form p=4k+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.

7. A prime number of the form p=3k+1 can be expressed as , where u and v are whole numbers.

8. A prime number which is either of the form p=8k+1 or p=8k+3 can be expressed as , where u and v are whole numbers.

9. 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.

10. Fix a prime number p, and consider the sequence of powers of 2:

    

    Then:

    1. There is always a smallest exponent n such that is a multiple of p.

    2. This n is always a divisor of p-1.

    3. If m is any other exponent such that is a multiple of p, then m is a multiple of n.

11. The statements above remain true when we replace 2 with any other number that is not divisible by our prime number p.

12. Given any number N that is not a square, there exist infinitely many pairs of whole numbers x and y such that .

13. In the previous problem, there exists an explicit ``mechanical'' procedure for finding all the solutions x and y.

14. No cube can be written as a sum of two cubes.

15. No fourth power can be written as a sum of two fourth powers.

16. In general, if n is bigger than 2, no n-th power can be written as the sum of two n-th powers.