Take any whole number $q$. Calculate $q^2-1$. Factorize $q^2-1$ to give two factors $a$ and $b$ (not necessarily $q+1$ and $q-1$). Put $c=a+b+2q$ . Then you will find that $\mathrm{ab}+1$ , $\mathrm{bc}+1$ and $\mathrm{ca}+1$ are all perfect squares. Prove that this method always gives three perfect squares.

The numbers $a_1,a_2,...a_n$ are called a Diophantine n-tuple if $a_r{a}_s+1$ is a perfect square whenever $r\ne s$ . The whole subject started with Diophantus of Alexandria who found that the rational numbers

${\displaystyle \frac{1}{16},\hspace{1em}\frac{33}{16},\hspace{1em}\frac{68}{16},\hspace{1em}\frac{105}{16}}$

have this property. (You should check this for yourself). Fermat was the first person to find a Diophantine 4-tuple with whole numbers, namely 1, 3, 8 and 120. Even now no Diophantine 5-tuple with whole numbers is known.