1. Prove that if is an odd prime, then is a difference of two squares in one (and only one) way.

2. In 1657, Fermat issued a mathematical challenge to John Wallis (1616-1703) and the rest of England: to find a cube, which, added to its proper divisors, makes a square. Wallis’ reply was that he was too busy to concern himself with such matters, and "in any case the number 1 is clearly a solution." William Brouncker then suggested the solution . Fermat was unimpressed with the non-integer solution, and asked for an integer solution. Find one.

3. Show that is composite if has an odd prime factor. (Hint: Let where is an odd integer, and show that has as a factor).

4. Thinking of Fermat’s method of proving, why do you think he only dealt with positive integers?

5. List four distinct mathematical fields in which Fermat played a part.