Section 14.3 Related Questions About Sums
There is yet another generalization that will serve better as a lead-in to the next chapters. Think about the following two problems.What numbers can be written as x^2+2y^2\text{?} (Think of it as x^2+y^2+y^2\text{.})
What numbers can be written as x^2+3y^2\text{?}
Fact 14.3.1.
No number
\begin{equation*}
n\equiv 5\text{ or }n\equiv 7\text{ (mod }8)
\end{equation*}
can be written as x^2+2y^2\text{.}
Proof.
Try all numbers modulo 8 and see what is possible! (See Exercise 14.4.3.)
\begin{equation*}
x^2+2y^2=(x-\sqrt{2}iy)(x+\sqrt{2}iy)
\end{equation*}
you could start proving such things. When might a square root of two exist modulo p β¦
Here are some numbers which can be written in this form.
xxxxxxxxxx
1
@interact
2
def _(n=10):
3
pretty_print(html("Using $a$ and $b$ up to $%s$:"%n))
4
L=[a^2+2*b^2 for a in [0..n] for b in [0..n]]
5
L.sort(); print(L)
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