NTIC Related Questions About Sums

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 $x^2+3y^2\text{?}$

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These are very natural generalizations to the “two squares” question. How could we approach them? Here's one type of idea.

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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 $x^2+2y^2\text{.}$

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Proof.

Try all numbers modulo 8 and see what is possible! (See Exercise 14.4.3.)

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Already Fermat (unsurprisingly) claimed a partial converse to Fact 14.3.1. He stated that any prime number

$p$ which satisfies

$p\equiv 1$ or

$p\equiv 3\text{ (mod }8)$ could be written as a sum of a square and twice a square.

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This time, Euler wasn't the one who proved it! But you could almost imagine that by factoring

$\begin{equation*} x^2+2y^2=(x-\sqrt{2}iy)(x+\sqrt{2}iy) \end{equation*}$

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you could start proving such things. When might a square root of two exist modulo

$p$ …

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Here are some numbers which can be written in this form.

xxxxxxxxxx
 
@interact
def _(n=10):
    pretty_print(html("Using $a$ and $b$ up to $%s$:"%n))
    L=[a^2+2*b^2 for a in [0..n] for b in [0..n]]
    L.sort(); print(L)

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In Exercise 14.4.12, you will try to discover a similar pattern for

$x^2+3y^2\text{.}$ See also Section 15.4.