Euler found four whole numbers such that the sum of any two of
the numbers is a perfect square. Three of the numbers that he
found are
a = 18530, b=65570, c=45986.
Find the fourth number, x. You could do this by trial and error
(sometimes called trial and improvement), and a spreadsheet would
be a good tool for such work. However, Euler would not have used
any electronic calculating aids to find his 'fearsome foursome'
and he would have found ways of reducing the search to a small
number of cases and this is what you should try to do. You could
do this by writing down
a+x = P2
b+x = Q2
c+x = R2,
and then focussing on Q2-R2=b-c which is known. Moreover you
know that Q > Öb and R > Öc . Use this to show that
Q-R £ 41 . Use a spreadsheet to calculate values of Q+R ,
Q and x for values of Q-R from 1 to 41 , and hence to find
the value of x for which a+x is a perfect square.
There may be better ways to do this, and if you find one,
do let us know!