Determine the integer solutions $(a, b)$ of

$$
\frac{1}{a}-\frac{1}{b}=\frac{1}{37}
$$

## Sophie Germain Factorization

## Proposition

Let $a, b$ be integers. Then,

$$
a^{4}+4 b^{4}=\left(a^{2}+2 b^{2}+2 a b\right)\left(a^{2}+2 b^{2}-2 a b\right)
$$

## Proof

It suffices to write $a^{4}+4 b^{4}=a^{4}+4 a^{2} b^{2}+4 b^{4}-4 a^{2} b^{2}=\left(a^{2}+2 b^{2}\right)^{2}-(2 a b)^{2}$.

Remark

We can observe, in particular, that $a^{2}+2 b^{2}-2 a b=(a-b)^{2}+b^{2} \geq b^{2}$.
Once again, the equation rewrites as $(37-a)(b+37)=37^{2}$. We then obtain several cases, among which only one solution has positive values: $a=36, b=36 \times 37=1332$.