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 + 2 q

. Then you will find that

ab + 1

bc + 1

and

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 $r \neq s$ . The whole subject started with Diophantus of Alexandria who found that the rational numbers

$\frac{1}{16}, \frac{33}{16}, \frac{68}{16}, \frac{105}{16}$

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.