**Goldbach Conjecture**. Christian Goldbach (1690-1764) conjectured that: **all number greater than 2 and even is equal to the sum of two primes.**

Here are some examples:

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

Also known as the **strong Goldbach conjecture** has been proven using computers to the even numbers up to one hundred million, but has not yet been demonstrated.

In contrast, the **Goldbach’s weak conjecture** which states that **any number greater than 5 and odd is equal to the sum of three primes** was recently demonstrated by the Peruvian mathematician **Harald Helfgott**.

**Fermat Conjecture**. Pierre de Fermat (1601-1665) conjectured in 1637 that **there are no integers verifying the following equation for n greater than or equal to 2**

Fermat’s conjecture remained unproven three centuries and a half. In June **1993**, the English mathematician **Andrew Wiles** announced that he had proved Fermat’s conjecture. But his show had some gaps, it took over a year to resolve. Finally, Fermat’s conjecture has been established, becoming the **Fermat’s last theorem**.

**Fermat’s little theorem**: 2 ways to express:

the other way is:

Example:

**Fermat’s theorem on sums of two squares** **Every prime number p can be written as**: , **where x and y are integers if p = 2 or p=1 (mod 4).**

{5,13,41} are of 4k + 1 form, or in other words are congruent to 1 mod 4.

**Euler’s Theorem** (1736) **If a and n are coprimes:**

in the language of the elements Zn is:

where is the **Euler’s totient**, which counts the **coprime** numbers with** n** **up to n**.

**Mills’ Constant (1947)**

The constant generates prime numbers for any n natural.

If the Riemann hypothesis is true the value of the constant is **= 1,30637788386308069046….**

since there is no known way to calculate this constant and it is unknown if the rational number.