NTIC Sums of Squares

Chapter 13 Sums of Squares

We have now more or less exhausted a lot of what we can do with linear questions, and even gone beyond to many nonlinear ones. With that in mind, we return to other considerations. As a warmup for this and ensuing chapters, consider the following question.

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Question 13.0.1.

Take a positive integer $n$ (or zero) and try to write it as $n = a^{2} + b^{2}$ for $a, b \in Z .$ For which $n$ is this possible, for which is it not?

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It seems that Albert Girard already knew the answer to this question in the first quarter of the 17th century, and Fermat discovered it a couple years later as well. A full proof of the answer to this question did not come until Euler (no surprise here) about six score years after that.

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Historical remark 13.0.2. Albert Girard.

Girard is an interesting figure, less well-known than his contemporaries. He apparently was the first to use our modern notation for trigonometric functions, and spent his adult life in the Netherlands escaping religious persecution as a Protestant in France.

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Historical remark 13.0.3. Leonhard Euler.

Euler is well known for being a rather conventional religious family man amidst the Enlightenment court of Frederick the Great, and for taking a lot of teasing from Voltaire and the king (among other things, for being partly blind at the time). See [E.5.6] for much more about him and his work¹ at the level of this text, or over one third of [E.5.8] for a detailed perspective by an eminent number theorist, or simply browse the Euler Archive.

There is an immense recent English-language biography about him, but I do not actually recommend it for most readers.

There is a lot more to say about someone universally acknowledged as one of the greatest mathematicians of all time, but we already have plenty of Euler's work in this book for you to peruse.

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Historical remark 13.0.4. Pierre de Fermat.

We've already seen Fermat's work several times (such as Subsubsection 3.4.3.2, Theorem 7.5.3, and Subsection 12.1.1), and we'll see another glimpse of him in Question 15.6.5. About the man himself we know less, mostly that he was a jurist in southern France who didn't travel much, but corresponded a fair amount about his mathematics, which included prototypes for both differential and integral calculus! As with most things about Fermat's personal life, it's less well known that he also had a religious side; in [E.7.12] a well-known classicist translates a moving poem about the dying Christ written in honor of one of Fermat's friends. See [E.5.8, Chapter II] for many mathematical, and some personal, details.

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So try out Question 13.0.1! Some things to think about while you try this:

Are any special types of numbers easier to write in this way than others?
Is there any way of generating new such numbers from old ones?
If some types of numbers are not a sum of squares, how might you prove this?

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A separate question to at least keep track of is this.

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Question 13.0.5.

Assuming you can indeed write it in this way, how many ways you can write a number as a sum of squares?

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This chapter is completely devoted to continuing to address questions about writing numbers as a sum of two squares. It will lead us a little far afield, of necessity, to ask (and start to answer) questions about congruences again. Much of this chapter will be devoted to a geometric proof that certain numbers are indeed representable as a sum of two squares. This chapter is a perfect illustration of one of the main themes of this text – the unity of mathematics.

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Summary: Sums of Squares

This chapter examines the question of what numbers may be written as a sum of two perfect (integer) squares.

First an exploration of the problem is in order, including a geometric interpretation and the famous identity Fact 13.1.7.
In Proposition 13.2.4 we show that prime numbers may essentially only be written in one way as such a sum.
Defining the square root of a number, modulo $n,$ is the content of Definition 13.3.1, which we then immediately use to find out when $- 1$ has a square root.
The proof of Theorem 13.4.5 that primes of the form $4 k + 1$ can be written as a sum of squares is a real geometric treat.
In the penultimate section we prove Theorem 13.5.5, which explains why even though $21$ can be written as $4 k + 1,$ it cannot be written as a sum of squares.
We finish with A One-Sentence Proof of the main theorem of this chapter.

The Exercises give practice filling in many of the smaller details of the proofs.