Hello, everyone!
Last week you learned basics on prime numbers.
This week you will learn laws of prime numbers.
More than 2,000 years ago Euclid proved there are infinitely many prime numbers.
The Prime Number Theorem proved by Hadamard and de la Vallée Poussin in 1896
says the number of prime numbers less than or equal to N is approximated by the function N / log(N).
We can improve the Prime Number Theorem if we knew the Riemann Hypothesis.
The Prime Number Theorem is a distribution law.
It is a law concerning the totality of prime numbers.
But it does not say anything about individual prime numbers.
There are many other laws.
Today it is known that each individual prime number also obeys beautiful laws called Reciprocity Laws.
Fermat was a lawyer from Toulouse, France in the 17th century.
He contributed to many areas of mathematics and physics
including number theory, geometry, analysis, and optics.
He is one of the pioneers of Reciprocity Laws.
He discovered many beautiful laws of prime numbers.
Perhaps Fermat is famous for a theorem now called Fermat's Last Theorem.
It says no positive integers X, Y, and Z satisfy the equation
X^N + Y^N = Z^N if N is greater than or equal to 3.
He claimed to prove it in posthumous writings.
But there is no evidence that he actually proved it.
Fermat's Last Theorem had been one of the most famous unsolved problems for more than 300 years.
It was finally proved by Wiles in the end of the 20th century.
I think it is a miraculous coincidence that Fermat was a pioneer of Reciprocity Laws of prime numbers,
and Wiles proved his last theorem by
establishing new Reciprocity Laws.
Wiles also used some methods from the theory of Kummer and Iwasawa.
The topic of this week's lecture is another theorem of Fermat,
called Fermat's Theorem on Sums of Two Squares.
This is an important theorem because it can be considered as the birth of the reciprocity laws.
The statement is simple.
A prime number P is a sum of two squares
if and only if P is 2 or P is 4N + 1 for some N.
Let me give you some examples.
For example, 13 = 4 * 3 + 1.
In fact, 13 can be written as a sum of two squares
because 13 = 2^2 + 3^2.
This is a surprising theorem because there is no simple relation between 4N + 1 and the sum of two squares.
This theorem was first observed by Girard in 1625.
It is not clear whether Fermat actually proved it
because he did not usually write a proof of his discoveries.
The first complete proof was given by Euler in 1740s.
Fermat's Theorem on Sums of Two Squares is an extremely influential theorem
because it shows a surprising connection between squares and prime numbers.
It is also the first non-trivial case of the Reciprocity Laws on prime numbers.
Since it is a beautiful theorem, many mathematicians have studied it.
Several different proofs are known today.
It is worth remarking that the uniqueness holds.
Namely, if a prime number P can be written as a sum of two squares in two different ways,
for example, X^2 + Y^2 = S^2 + T^2,
then the pair (X, Y) should be equal to (S, T) or (T, S).
It is interesting to see that Fermat's Theorem on Sums of Two Squares is not true for non-prime numbers.
For example, 21 = 4 * 5 + 1.
But 21 cannot be written as a sum of two squares.
65 is written as the sum of two squares in two different ways.
21 and 65 are not prime numbers.
This theorem shows deep properties of prime numbers.
Many different proofs of Fermat's Theorem of Sum of Two Squares are known today.
Some of them are elementary,
some of them require advanced techniques.
In 1990 Zagier found an extremely short proof of this theorem.
His proof consists of one sentence only.
Here is Zagier's proof.
It is true that the essential part of his proof is just one-sentence long.
But it's not easy to understand because his proof is based on an ingenious counting argument.
