Introduction - Elliptic Curves: Number Theory and Cryptography - page 1

Chapter 1

Introduction

Suppose a collection of cannonballs is piled in a square pyramid with one ball
on the top layer, four on the second layer, nine on the third layer, etc. If the
pile collapses, is it possible to rearrange the balls into a square array?

Figure 1.1
A Pyramid of Cannonballs

If the pyramid has three layers, then this cannot be done since there are
1 + 4 + 9 = 14 balls, which is not a perfect square. Of course, if there is only
one ball, it forms a height one pyramid and also a one-by-one square. If there
are no cannonballs, we have a height zero pyramid and a zero-by-zero square.
Besides theses trivial cases, are there any others? We propose to find another
example, using a method that goes back to Diophantus (around 250 A.D.).
If the pyramid has height x , then there are

+ x 2 = x ( x + 1)(2 x +1)
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1 2 +2 2 +3 2 +

···

balls (see Exercise 1.1). We want this to be a perfect square, which means
that we want to find a solution to

y 2 = x ( x + 1)(2 x +1)
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