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 $x^{2} + 2 y^{2} ?$ (Think of it as $x^{2} + y^{2} + y^{2} .$ )
What numbers can be written as $x^{2} + 3 y^{2} ?$

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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

n \equiv 5 or n \equiv 7 (mod 8)

can be written as $x^{2} + 2 y^{2} .$

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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

p \equiv 3 (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

x^{2} + 2 y^{2} = (x - \sqrt{2} i y) (x + \sqrt{2} i y)

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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} + 3 y^{2} .