As the text discusses, Plato devised an algorithm for finding Pythagorean Triples. This algorithm corresponds to the following formula, where $\displaystyle {p}>{1}.$:

$\displaystyle {\left({p}^{{2}}+{1}\right)}^{{2}}={\left({2}{p}\right)}^{{2}}+{\left({p}^{{2}}-{1}\right)}^{{2}}$

For each of the following values of $\displaystyle {p},$ find the three numbers that belong to that Pythagorean Triple. Write your answers as a list, separated with commas. For example, to specify the triple $\displaystyle {3},{4},{5},$ you would enter 3,4,5

(A) If $\displaystyle {p}=$ 21:
The numbers in the triple are:

(B) If $\displaystyle {p}=$ 46:
The numbers in the triple are:

(C) If $\displaystyle {p}=$ 34:
The numbers in the triple are:

Plato

(Source: http://www.unc.edu/courses/2005fall/phil/024/001/