Diophantus of Alexandria/Arithmetica/Book 3

Problems by Diophantus of Alexandria: Arithmetica Book $\text {III}$

Problem $6$

To find $3$ numbers such that their sum is a square and the sum of any pair of them is a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p + q + r$ is square

$p + q$ is square

$q + r$ is square

$r + p$ is square.

What are those $3$ numbers?

Problem $12$

To find $3$ numbers such that the product of any $2$ of them added to the $3$rd gives a square.

That is, let $\set {p, q, r}$ be a set of $3$ natural numbers such that:

$p q + r$ is square

$q r + p$ is square

$r p + q$ is square

What are those $3$ numbers?