“Read Euler, read Euler, he is the master of us all” written by Robin Wilson or “Euler: the master of us all” written by William Dunham are to show us how great and multifaceted Euler was as a mathematician. Indeed, he was. In this post, I want to write how great he was as an educator.

Today was my first session of a course in number theory. I had to start with “sums of two squares“. Like most other things in introductory number theory, this one also starts with a very simple observation. Some numbers like 5 ( \( 2^2+1^2 \) ) can be written as a sum of two squares, and some like 7 cannot be written as such. The natural question is which number is which. The students I had to work with had studied some number theory in the previous semester, but not primitive roots and quadratic residues, and of course, not the Gaussian integers. Basically, they hadn’t got the tools that are usually used to study (i.e., to prove useful facts about) the problem of our interest. Fortunately, Euler was there with two amazingly exploratory and yet rigorous enough pieces of work (as ever) on the subject: first work published in 1758, “On numbers which are the sum of two squares“, and the second work published in 1760, “Proof of Fermat’s theorem that every prime number of the form \(4n+1\) is the sum of two squares“. The two works are lengthy by today’s standard where using the “right” tools, there is even “a one-sentence proof” (due to Zagier) for what Euler proves in the first 12 pages of the first paper (its English version is 24 pages) plus the second paper (that is 11 pages). You can re-write all those 23 pages in just one page, calling it Euler’s proof by infinite descent. By doing so, you would have a nice and presentable proof of Fermat’s theorem (for an elementary number theory course as mine). However, you would miss most of the mathematical insights that your students (that mathematically might be as naïve as mine) could gain from the original text. A simple example of such insights is the mere fact that Euler calls his first argument (given in 1758) “Attempt at a Proof” not “A Proof”. This distinction would be just one of the lessons for your students. There would be many of such lessons as I try to convince you now. First, a few words about my plan is in order since it is quite related to what we might get from Euler.

The first step was to ask the students to list “those numbers which arise from the sums of two squares” (Paper I), say up to 50 (Euler himself listed them up to 200 without any use of calculators, mobiles, computers and so on!). The next step was to let them see and come up with any general statement that might be true for all such numbers, or for the numbers which cannot be written as the sum of two squares. I believed whatever they come up with could be found somewhere in Euler’s texts, perhaps not the exact things, but for sure, something quite related. Thus, after examining the statements found in the class, I could direct them to the Euler’s text where he addresses the same statement or something similar. It was the plan and it went better than I expected.

The first observation* was that powers of 2 can be written as a sum of two squares:

\( 2=1^2+1^2, 4=2^2+0^2, 8=2^2+2^2, 16=4^2+0^2, … \)

We called our first observation Theorem 1.

**Theorem 1**: Any power of two can be written as a sum of two squares.

The students had some experience in “university mathematics”, but most of them felt no need to prove our first theorem since it was true for all the examples on the blackboard! Here again, Euler comes in rescue, saying “in this class (dissertation) of many such statements (propositions), which until now have been accepted without proofs, we (I) will furnish proofs of their truth” (Paper I). Thus, I asked students to prove Theorem 1. To my surprise, they chose to prove it by mathematical induction (perhaps, because they had a lot of such proofs in the first semester). This is the way they did it:

Base: \( 2=1^2+1^2 \)

Assume that \( 2^k \) can be written as a sum of two squares.

\( 2^(k+1)=2. 2^k \) . Thus,\( 2^(k+1) \)can be written as a sum of two squares.

It was not easy for students to see why the product (here, \( 2^(k+1) \) does not automatically inherit the property of its factors (here, the factors are 2 and \( 2^k \) and the property is being a sum of two squares). Again Euler has something to say about this inheriting phenomenon. For example, a number that is a sum of two squares but neither of its factors is a sum of two squares. I used a silly example, giving that 3 and 5 are prime numbers (the property here is being a prime) but their product is not a prime number! This discussion put forward three options:

(i) Ignore proving Theorem 1 (that indeed wasn’t an option)

(ii) Choose a different direction to prove it.

(iii) Amend our failed proof.

The students chose the last option and this brought us to our next theorem.

**Theorem 2**: If \( m \) and \( n \) are two numbers, each of which is the sum of two squares, then their product \( mn\) will also be the sum of two squares.

Interesting, Euler suggests a number of simpler propositions that are special cases of Theorem 2 before giving the general form, because by it (special cases) this (the general case) will be more easily observed (Paper I). In fact, one of his lemmas (special cases) was enough to complete our proof by induction: If a number \( m \) is a sum of two squares, then so will be \( 2m \). However, we decided to proceed by proving the general case. Then, in addition to correcting our mathematical induction, we wrote 65 and 1105 (both suggested by Euler) as sums of two squares, using our proof of Theorem 2.

Euler has it all. He plays with examples, draw conclusions, warn you not to rely on them, seek proofs, guess when you might think wrong or overgeneralize, discuss them, give counterexamples and so on.

I haven’t yet distributed his papers in the class. This post was just about the first session. I’ll complete this story.

*Euler himself considered the square numbers first.

Theorem 2 had many lessons for us. Practically, we now knew how to write a number as sums of two squares if it was the product of sums of two squares. More importantly, theoretically, we realize that the factors of a number are important. Thus, we started to play with this idea and see if we can say anything about the number based on its factors and vice versa. Here again, Euler knew what would be going on in the class! For example, he warns us against the converse of Theorem 2, something that many students took the same as the theorem itself. Interestingly, to argue that it is correct they were saying “it is just the converse.” Anyway, we played with these ideas, conjectured one after the other and proved one after the other. Obviously, our conjectures were more than our proofs. Here are some of the conjectures that were named after the first student who suggested that:

Dan’s conjecture: If two numbers are sums of two squares then their division is also a sum of two squares. (Reformulated in terms of multiplication and explored as a homework problem; see Euler’s Proposition I)

Bek’s conjecture: [Prime] numbers in the form of 4n+3 cannot be written as a sum of two squares. (Proved by the analysis of the remainders)

Oli’s conjecture: Suppose a number has a prime factor in the form 4n+3, say p. If the number of times that p appears is even, the number can be written as a sum of two squares. (Proved using Theorem 1, Theorem 2, and assuming the correctness of Tuy’s conjecture).

Tuy’s conjecture: Primes in the form of 4n+1 can be written as sums of two squares.

At this point, I distributed Euler’s papers highlighting the fact that Euler has referred to the “proof” given in the first paper as “Attempt at a proof”. Consider that the point of the class was not to prove the theorem of sums of two square completely.

There were some other smaller conjectures on the way. Also, I gave some for homework. For the final exam, students will read part of Euler’s original text and will answer some questions asked about the way he has proved one of his theorems. Our exam will be next week, thus, I cannot give you more details now 🙂 But, I’ll do as soon as possible.

What happened in the subsequent session(s) ? And when you distributed Euler’s papers?