Finding a positive integer that can't be expressed in a certain form

Question

I attended a math speech and the speaker left the following question as an exercise:

Which positive integer cannot be expressed in the form $$x^2+2y^2+5z^2+5w^2?$$

I've trying to solve it but I haven't accomplished it yet. Any help is welcome

Answer 1

There are very few integers that are perfect squares, and none of them are negative. So we can order them starting from the smallest one, as $0$, $1$, $4$, $9$, $16$, $25$, etcetera. Now for any given positive integer $n$, there are only finitely many candidates for the squares $x^2$, $y^2$, $z^2$ and $w^2$ because all the coefficients are positive. Simply check them all to see whether $n$ can be expressed in this way.

Of course there is seemingly no guarantee that you will ever find a positive integer $n$ that is not of this form, even if it exists. But there is a wonderfully surprising theorem, the fifteen theorem, that states that if every positive integer up to $15$ can be expressed as such a sum of squares, then every positive integer can be expressed as such a sum of squares. So the approach described above only requires you to check up to $n=15$ to find a positive integer not of this form, if it exists.

A quick check shows that every integer $n<15$ is of this form, but that $n=15$ is not. A more precise version of the fifteen theorem then tells us that every integer $n>15$ is also of this form, thanks to lulu's comment with this link. So $n=15$ is the unique positive integer not of this form.