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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.
Diophantine N-tuples
Age 14 to 16
Challenge Level 
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.
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.
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