The 3-square theorem<br>

It is a famous theorem of Lagrange that every positive integer
is a sum of 4 perfect squares (some of which may be 0), but more subtle
is the case of 3 squares because not every positive integer can be
expressed in this form.  For example, 7, 15, and more generally numbers
of the form 8k+7 cannot be written in this way.  It is a hard theorem
of Legendre that classifies exactly which integers are a sum of 3 perfect squares.
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Often in number theory the problem of studying integer solutions to an equation lies
much deeper than the study of rational solutions, as the case of rational solutions
is often more susceptible to geometric methods.  Incredibly, it turns out that
if an integer is a sum of 3 rational squares then it is automatically also a sum
of 3 perfect (integral) squares!  After discussing a little history,
we will show the nifty argument that proves
this fact (which can be used as a step toward a simplified proof of Legendre's theorem).