Introduction

Abstract

What do the relations

$$ {5^2} = {3^2} + {4^2} $$

((i))

$$ 6 = {1^2} + {1^2} + {1^2} + {1^2} + {1^2} + {1^2} = {2^2} + {1^2} + {1^2} $$

((ii))

$$ 7 \ne {a^2} + {b^2} + {c^2} $$

((iii))

have in common? Obviously, their right hand members are all sums of squares. One way to describe those relations is as follows:

• i The square 5² can be represented, in essentially one way only, as the sum of two squares,

• ii The integer 6 can be represented in (at least) two essentially distinct ways as a sum of squares.

• iii The integer 7 cannot be represented as a sum of three squares.