Number Theory is the study of the properties of integers. A great part of these properties are concerning prime numbers. However, there are also studies about rational numbers, irrational numbers, transcendental numbers, Diophantine equations etc.

Since we have already talked about Fermat’s other works, we will just exemplarily show a few of his theorems and discoveries in number theory, and, wherever possible, the proofs for those theorems (most of them are copied down from other pages since they all look quite the same in general).
  

Diophantine Equations
  

Diophantine equations are equations with integral solutions.
Fermat’s Little Theorem is a famous example of this kind of equation:

If p is prime and a is relatively prime to p, 
then   a^{p - 1} – 1 divisible by p,  
i.e.   a^{p –1}   1 (mod p).

Proof (by Leonhard Euler):

Start by listing the first p – 1 positive multiples of a:

a, 2a, 3a, ... (p – 1)a.

Suppose that ra and sa are the same modulo p, then we have r s (mod p), so the p – 1 multiples of a above are distinct and  nonzero; i.e.  they must be  congruent to 1, 2, 3, ..., p – 1 in some order.

Multiply all these congruences together and we find

a.2a.3a. ... .(p –1)a 1.2.3. ... .(p – 1) (mod p),

or better, a(p – 1)(p – 1)! (p – 1)! (mod p).

Divide both sides by (p – 1)! to complete the proof.

 

Fermat’s Little Theorem is equivalent to the following

Corollary

Let p be a prime and a any integer, then ap a (mod p).

Proof

The result is trival (both sides are zero) if p divides a. If p does not divide a, then we need only multiply the congruence in Fermat’s Little Theorem by a to complete the proof.
 

Note

Fermat found his Little Theorem through his investigations of Euclid’s theorem on perfect numbers, namely through his investigation of the primality of 2^p – 1.

  Infinite Descent

 
The Method of Infinite Descent basically uses a contradiction argument. Suppose we want to show that there is no positive integer with certain properties. Now firstly, we assume there is such a number a. We then deduce that there exists a number b with b < a  such that b also possesses these properties.  Repeating this argument, we obtain an infinite descent in all the integers.  This contradicts the fact that there must be a smallest integer with these properties.

Note

Fermat found this method useful for many problems which were unsolved for many years. He actively used this method for proofs of his propositions. e.g. he used this method to prove the statement:

the area of a right triangle cannot be a square.  (i.e. there are no integers x, y, z satisfying the equation     x⁴ + y⁴ = z².   )

Proof

1.   Assume there are such integers.

2.   Then since x², y² and z² are Pythagorean triples, we can write them as follows:
 

y² = 2pq

x² = p² – q²

z = p² + q²

3.   Since 2pq is a square  then either p is even or q is even, so p² can be written as a Pythagorean triple again:
 

x² + q² = p²

where x, p and q are the Pythagorean triples and can be written as follows:
 
 

q = 2rs

x² = r² – s²

p = r² + s²
 

4.   Since        y² = 2pq            q  =  2(u²)      and     
                      p  =  v²               2u² = 2rs

5. If this is true, then   r = g²  and   s = h²

6. Substituting r and s as above, and using  p = v², we get the derived equation
 

g⁴ + h⁴ = v²           where           v < z.

7. this process is infinitely repeatable      contradiction.
 

Examples

Below are some problems for which Fermat found the solution using the infinite descent method:

• The sum of two cubes is never a cube.

• (2²)ⁿ + 1 is a prime number.

• There is only one solution to the equation   x² + 2 = y³.

• 2(x²) – 1 = p,  2(y²) – 1 = p²  have only integral solutions for  p = 1 or 7.
 

Fermat Numbers 

Fermat Numbers are integers of the form:

, where n is a non-negative integer.

Since F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537, Fermat induced that all of these numbers are primes, but later Euler proved ingeniously that 641 divides F₅. In fact, for n between 4 and 31, F_n is a composite number.

A quick means of testing the primality of F_n is via Pepin’s Test:
 

F_n is prime       ³/₂.(F_n – 1) –1 (mod F_n).

 
Here you can find some Fermat Numbers and their factoring status.

Fermat Polygonal Number Theorem
 

Every positive integer is a sum of at most 3 triangular numbers, 4 square numbers, 4 pentagonal numbers, and n N-gonal numbers.

Proofs

Here is a summary of progress made in the proof of this theorem.

Triangular case   –  Gauss 1796
Identity               –   Euler identity
Square case        –   Lagrange 1772 (using Euler identity)
                           –   Jacobi
General case       –   Cauchy 1813

We do not show the proofs here, since they are quite complicated, but the interested reader can find them easily on the World Wide Web.