Take any whole number q. Calculate q2 − 1. Factorize q2 − 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 ab + 1 , bc + 1 and
ca + 1 are all perfect squares. Prove that this method always gives
three perfect squares.
The numbers a1, a2, ... an are called a Diophantine n-tuple if
aras + 1
is a perfect square whenever r ≠ s . The whole subject started
with Diophantus of Alexandria who found that the rational numbers
1
16
,
33
16
,
68
16
,
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.