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## 'Diophantine N-tuples' printed from http://nrich.maths.org/

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 $ 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_ra_s + 1 $ is a perfect square whenever $ r \neq s $ . The
whole subject started with Diophantus of Alexandria who found that
the rational numbers $$ {1 \over 16},\ {33\over 16},\ {68\over
16},\ {105\over 16} $$

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.