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.