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Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.


The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

Lower Bound

Age 14 to 16
Challenge Level

Investigate the following sequence of fraction sums:
\begin{eqnarray} \frac{1}{2} &+& \frac{2}{1} = \\ \frac{2}{3} &+& \frac{3}{2} = \\ \frac{3}{4} &+& \frac{4}{3} = \\ \frac{4}{5} &+& \frac{5}{4} = \end{eqnarray}
What would you get if you continued this sequence for ever?

What do you think will happen if you add the squares of these fractions, that is:
\begin{eqnarray} \left(\frac{1}{2}\right)^2 &+& \left(\frac{2}{1}\right)^2 = \\ \left(\frac{2}{3}\right)^2 &+& \left(\frac{3}{2}\right)^2 = \end{eqnarray}
and so on?