In Chapter 1 we saw that the nth triangular number Tn is given by the formulaThe first few...

In Chapter 1 we saw that the nth triangular number Tn is given by the formula

The first few triangular numbers are 1, 3, 6, and 10. In the list of the first few Pythagorean triples (a, b, c), we find (3, 4, 5), (5, 12, 13), (7, 24, 25), and (9, 40, 41). Notice that in each case, the value of b is four times a triangular number.

(a) Find a primitive Pythagorean triple (a, b, c) with b = 4T5. Do the same for b = 4T6 and for b = 4T7.

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(b) Do you think that for every triangular number Tn, there is a primitive Pythagorean triple (a, b, c) with b = 4Tn? If you believe that this is true, then prove it. Otherwise, find some triangular number for which it is not true.