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, and a spreadsheet would be a good tool
for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then
focus on Q^2-R^2=b-c which is known. Moreover you know that Q >
sqrtb and R > sqrtc . Use this to show that Q-R is less than or
equal to 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.
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 a1, a2, ... an are called a Diophantine n-tuple if aras + 1 is a perfect square whenever r is not equal to s . The whole subject started with Diophantus of Alexandria who found that the rational numbers
1/16, 33/16, 68/16 and 105/16 have this property. 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.
Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you
notice when successive terms are taken? What happens to the terms
if the fraction goes on indefinitely?
You may find the Excel file Data Chunks useful.
If you spend a moment looking at the numbers you'll soon see how
this spreadsheet file works.
There is also something you should know about spreadsheets and
Using ICT is often brilliant for getting lots of results fast,
leaving your mind free to think about what's going on, but doing
some calculating yourself gives you an on-the-ground feel for the
So the trick is to use both approaches, getting the benefit from
The Data Chunks problem is a challenge.
It takes time and determination, but if you've enjoyed wrestling
with it then we feel confident that you'll want to see these links
There is an NRICH article by Alan and Toni Beardon about
Click for Part One
then there's a Part Two
to take you on further.
Another article, this time by Vicky Neale and Matthew Buckley is
Yet another by Vicky is called Introductory