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### Construction of regular polygons

There are a few classic problems of ancient mathematics that are easy to state but incredibly difficult to solve. Take for example, the problem of constructing regular polygons and the problem of trisecting an angle by compass and straightedge. It was not until the 18th-19th centuries that mathematicians could finally solve them by employing advanced tools of number theory and algebra. Today, we will look at the regular polygon construction problem.
Regular polygon construction problem. Using compass and straightedge, construct a regular polygon with $n$ sides.

First of all, we note that equilateral triangle and square are the two figures that can be easily constructed. We will call a polygon with $n$ sides an $n$-gon and make the following observation

Observation: If we can construct a regular $n$-gon then we can also construct a regular $2n$-gon.

Why is that? That is because if we have a regular $n$-gon then we can construct its circumcircle, and then at each side, we construct the perpendicular bisector to bisect the corresponding circle arc, then we will have a regular $2n$-gon!

With the above simple observation, from a regular $3$-gon, we can construct regular $6$-gons, regular $12$-gons, regular $24$-gons, etc.
 triangle $\rightarrow$ hexagon $\rightarrow$ dodecagon

And from "$2$-gon", we can construct regular $4$-gons, regular $8$-gons, regular $16$-gons, etc.
 2-gon $\rightarrow$ square $\rightarrow$ octagon

Thus, we have reduced the problem to the case when $n$ is an odd number. For instance, if we want to construct a regular polygon with $68$ sides, then we only need to find a way to construct a regular polygon with $17$ sides. This is because $68 = 17 \times 2 \times 2$, so from $17$-gon, we obtain $34$-gon, and then $68$-gon!

Problem. For an odd number $n$, construct a regular polygon with $n$ sides.

This turns out to be an extremely hard problem!

For a long time, we know how to construct a regular pentagon. In his glorious books, the Elements, Euclid of ancient Greek presented a method of constructing a regular pentagon by compass and straightedge. But for the next two thousand years, no one had ever succeeded in constructing a regular $7$-gon, a regular $9$-gon, or a regular $11$-gon. Every effort seems to end up in failure.

The reason is not because of our incapability but it turns out that it is impossible to construct those polygons!

The first mathematician that made a breakthrough is Gauss. Gauss is often referred to as "the Prince of Mathematics". His Disquisitiones Arithmeticae, published when he was 24 years old, stands to this day as one of the most important book in mathematics.

When he was 19 years old, Gauss made a major discovery in mathematics - the construction of a regular $17$-gon. It is believed that the excitement of this discovery had led Gauss to choose mathematics instead of philosophy as a career. Gauss was so pleased by this result that he requested that a regular $17$-gon be inscribed on his tombstone. For some unknown reason, this request was not fulfilled, but on Gauss's memorial stone in Brunswick we can find a 17-point star!

This is Gauss' ground-breaking theorem:
Gauss' Theorem. If $n = p_1 \dots p_t$ where $p_1$, ..., $p_t$ are distinct Fermat primes then a regular $n$-gon can be constructed by compass and straightedge.

Later in 1837, Wantzel, a French mathematician, established the opposite direction of the theorem, that is, for an odd number $n$, if a regular $n$-gon is constructible then $n$ must have the form $n = p_1 \dots p_t$ as above.

So, what are the Fermat's primes mentioned in Gauss' theorem?

Fermat's primes

Most of us probably have heard about the Fermat's last problem

The Fermat problem. Prove that for any $n \geq 3$, the following equation has no non-zero solutions $$x^n + y^n = z^n.$$

The Fermat's last problem had frustrated generations of mathematicians. It had attracted so much effort from top mathematicians as well as young school students. This is probably because it is stated so simple and neat. But the most likely reason is that it is a "mathematical mystery". Fermat wrote this problem on the margin of a book. He wrote that he had found a very beautiful proof but couldn't write it down because "there wasn't enough space" on the margin!

In 1994, Andrew Wiles - an English mathematician finally solved it. The proof employs some of the most powerful tools of modern mathematics. So Fermat's story is still a mystery!

Fermat was actually not a professional mathematician. He was a lawyer and he was doing math probably just for fun. He often communicated his results to other mathematicians of his time. "Ninety-nine times out of a hundred" Fermat was correct in his mathematical discoveries, but there was one time when he was wrong! This is about the Fermat's primes that we are about to discuss.

Since prime numbers serve as the building blocks for the entire natural numbers, mathematicians are keen on finding properties and formulas for primes. Fermat predicted that the numbers of the form $$F_n = 2^{2^n}+1$$ are prime numbers. If this was true then it would be a very nice formula for prime numbers. But unfortunately, this conjecture is not correct.

• $n=0$, $F_0 = 2^{2^0}+1 = 2^1 + 1 = 3$ is a prime,
• $n=1$, $F_1 = 2^{2^1}+1 = 2^2 + 1 = 5$ is a prime,
• $n=2$, $F_2 = 2^{2^2}+1 = 2^4 + 1 = 17$ is a prime,
• $n=3$, $F_3 = 2^{2^3}+1 = 2^8 + 1 = 257$ is a prime,
• $n=4$, $F_4 = 2^{2^4}+1 = 2^{16} + 1 = 65537$ is a prime,
• $n=5$, $F_5 = 2^{2^5}+1 = 2^{32} + 1 = 4294967297$ is a composite number!

The mathematician Euler pointed out that $F_5 = 2^{2^5}+1$ is a composite number as $$F_5 = 2^{2^5}+1 = 641 \times 6700417.$$

We have the following definition.
Fermat's prime. A prime number is called a Fermat's prime if it has the form $$F_n = 2^{2^n}+1.$$

Gauss-Wantzel Theorem

Going back to the problem of constructing regular polygons, the answer for the problem is the following theorem

Gauss-Wantzel's Theorem. For an odd number $n$, a regular polygon of $n$ sides can be constructed by compass and straightedge if and only if $n = p_1 \dots p_t$, where $p_1$, ..., $p_t$ are distinct Fermat's primes.

Since $3$, $5$, $17$, $257$, $65537$ are Fermat's primes, by the theorem, regular $3$-gons, regular $5$-gons, regular $17$-gons, regular $257$-gons, and regular $65537$-gons are constructible. In addition,
• $3 \times 5 = 15$ so regular $15$-gons are constructible,
• $3 \times 17 = 51$ so regular $51$-gons are constructible, etc...

On the other hand,
• regular $7$-gons are not constructible,
• $9 = 3 \times 3$ so regular $9$-gons are not constructible,
• regular $11$-gons are not constructible,
• regular $13$-gons are not constructible, etc...

With the polygons that are constructible, how can they be constructed? For instance, how can we construct a regular polygon with $15$ sides or $17$ sides? With the polygons that cannot be constructed, is there a method to construct them approximately with a small error?

There are many interesting questions need to be answered. However, we will save it for our next posts. Let us stop here for now. See you next time.

Homework.

Note that for the following problems, some of them have solutions, but some of them are open problems. However, we will not reveal it here and encourage readers to solve them all!

1. Prove that if $2^a + 1$ is a prime then either $a=0$ or $a=2^n$.

2. Prove that there are infinitely many primes of the form $2^a + 1$.

3. Prove that there are infinitely many composite numbers of the form $2^a + 1$.

4. Prove that for any $n > 1$, the last digit of the Fermat number $F_n = 2^{2^n}+1$ is 7.

5. Prove that there are infinitely many numbers $n$ such that the Fermat number $F_n = 2^{2^n}+1$ is a composite number.

6. Prove that for any $n > 1$, the Fermat number $F_n = 2^{2^n}+1$ cannot be written as a sum of two primes.

7. Find all pairs of numbers $n$ and $m$ such that $F_n F_m$ is a perfect square.

8. Prove that for any $n$, the Fermat number $F_n = 2^{2^n}+1$ is not a cubed number.

9. Prove that the Fermat number $F_{2014} = 2^{2^{2014}}+1$ is a composite number.