Fibonacci and Square Numbers - Questions for Investigation

1.  Find all the ways to express 225 as a sum of consecutive odd integers. Use your results to find the squares that can be added to 225 to produce another square.  What determines the number of ways in which a given number can be expressed as a sum of consecutive odd numbers? 

2.  Show that 336 is a congruous number.  Use your results to find a rational number x such that x² – 21 and x² + 21 are both squares of rational numbers.  Can you find examples with numbers other than 5 (shown in the text) and 21?

3.  There is a correspondence between ordered triples (a, b, c) with a² + b² = c² and ordered triples (p, q, r) with p², q², r² forming an arithmetic progression.  The triple (a, b, c) = (3, 4, 5) corresponds to (p, q, r) = (1, 5, 7), the triple (a, b, c) = (5, 12, 13) corresponds to (p, q, r) = (7, 13, 17), and the triple (a, b, c) = (8, 15, 17) corresponds to (p, q, r) = (7, 17, 23).

Discover the rule for this correspondence and explain why it works.

4.  Triangular numbers can be found by the taking the sum of all integers from 1 to n, so we get 1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, and so on.  Adapt as many of Leonardo’s results as you can to the case of triangular numbers.