'Given any pair of positive integers p and with p >4 prove that the triple of integers [a, b,c] where W = b = 2pq and c = p +9 form Pythagorean triple _ Why is the restriction p > q necessary?'
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Pythagorean triples A triple of positive integers $a, b,$ and $c$ is called a Pythagorean triple if $a^{2}+b^{2}=c^{2} .$ Let $a$ be an odd positive integer and let $$ b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left\lceil\frac{a^{2}}{2}\right\rceil $$ be, respectively, the integer floor and ceiling for $a^{2} / 2$ a. Show that $a^{2}+b^{2}=c^{2} .$ (Hint: Let $a=2 n+1$ and express $b$ and $c$ in terms of $n .$ b. By direct calculation, or by appealing to the accompanying figure, find $$ \lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil} $$
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