Henry Ernest Dudeney/Puzzles and Curious Problems/168 - Mental Arithmetic/Solution

by : $168$

 * Mental Arithmetic

Solution
gives $3$ and $5$:


 * $3^2 + 3 \times 5 + 5^2 = 49 = 7^2$

but we also have:
 * $7^2 + 7 \times 8 + 8^2 = 169 = 13^2$

Proof
Let $a$ and $b$ be the numbers in question.

We have that:

$m$ can be any integer greater than $1$, and $a$ is chosen to make $b$ rational.

The general values are:

Setting $m = 2$, we have:

Setting $m = 3$, we have:

and the two solutions are apparent.