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# 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.