A prime number is a positive integer, not equal to one,

and divisible only by itself, and by one. Examples are

2, 3, 5, 7, 11, 13, 17, …

The integer one will not be included because it would

disrespect the Fundamental Theorem of Arithmetic

which states that all integers greater than one factor

uniquely, up to reordering, into prime numbers.

Concerning the prime numbers, there exist interesting

theorems which have been proven rigorously and even

more interesting easy-to-state **unproven conjectures**

which could become theorems in our lifetimes.

* Infinite number of primes. (Theorem).

Proof: Assume there is a finite number of prime numbers

p_{l}, p_{2}, p_{3}, … p_{n}.

Take the product, of all n primes, and add one.

This is either a prime, or divisible by a prime, bigger than p_{n}.

So the assumption is false.

Therefore, there is an infinite number of primes.

This proof is in Euclid’s *Elements*, c. 300 BC.

* Vinogradov. (Theorem). [1937]

Every, sufficiently large, odd integer can be written

as the sum of three prime numbers.

* Chen. (Theorem).

Every sufficiently large even integer can be written

as the sum of a prime number, and a semiprime, where a

semiprime is a product of two prime numbers.

* Chen. (Theorem). [1966]

There is an infinite number of pairs of positive integers

(p, p+2) where p is prime or semiprime.

* Green-Tao. (Theorem). [2004]

The sequence of prime numbers contains arbitrarily long

arithmetic progressions. For any natural number k, there

exist k-term arithmetic progessions of primes.

* Dirichlet. (Theorem).

Every arithmetic progression, of the form (an+b), where a, n

and b are positive integers, contains an infinite number, of

prime numbers.

* Zhang. (Theorem) [2013]

The gap (p_{n+1} – p_{n}) between successive prime numbers

possesses a finite limit for asymptotically large n. The limit Zhang obtained was 70,000,000, subsequently lessened by others.

This can be regarded as progress toward proof of the twin-prime conjecture.

*** Legendre conjecture (Unproven) **

There always exists at least one prime between successive

perfect squares n^{2} and (n + 1)^{2}, where n = 1, 2, 3,…

*** Near square conjecture (Unproven) **

There are infinitely many primes p such that (p – 1)

is a perfect square; in other words, there are infinitely

many primes of the form n^{2} + 1.

*** Twin prime conjecture (Unproven) **

Made by de Polignac in 1849.

The number of pairs of primes (p_{1}, p_{2}), with

p_{2} = (p_{1} + 2), is infinite.

*** Generalized twin prime conjecture (Unproven) **

Also made by de Polignac in 1849.

The number of pairs of primes (p_{1}, p_{2}), such that

p_{2} = (p_{1} + n) with n any even integer, is infinite.

*** Goldbach conjecture (Unproven)** — exceptionally simple to state.

Every even integer, greater than two, can be written as the sum of two primes. This conjecture has stood unproven since 1742 when Goldbach made it in a letter to Euler who replied that it is definitely a theorem but he was unable to provide a proof, an inability easy to understand because it remains unproven 276 years later.