Homework Sheet Number One

1. Adapt the Pythagorean proof given by Aristotle on page 75 of our text to prove that the square root of 3 is not a rational number. Did you know that when we say a person is acting irrationally, we are referring to the root word for a number which can’t be expressed as a ratio of two integers?

2. (For math credit) Prove that the nth triangular number is n(n+1)/2.

3. Show, by a Pythagorean argument using arrays of points, that the nth square number is a sum of two triangular numbers.

4. (For math credit) Prove that the nth pentagonal number is equal to the nth square number plus the (n-1)st triangular number.

5. a. Show that 496 is a perfect number

b. Show that 220 and 284 are amicable numbers.