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Bang's Theorem

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Rudolff's Problem

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I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius 4 cm and touches one side of the box, the middle sized one has radius 9 cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

Diophantine N-tuples

Stage: 4 Challenge Level: Challenge Level:1

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.