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Interesting facts/conjectures:

1/ More than 2000 years, Euclid proved that there exists no largest prime.

2/A 17^th century monk, Mersenne found that number of the form 2ⁿ – 1 are primes when n is a prime.

3/In 1772, Euler   devised a formula n² – n + 41 which yields a prime for all natural numbers n up to 40, but fails for 41.

4/ In 1879, Escott devised a formula n² – 79n + 1601 which yields a prime for all natural numbers up to 79, but fails for 80.

5/In 1742, Goldbach conjectured in a letter to Euler that every even number greater than or equal to 6 can be expressed as a sum of two primes, and every odd number greater than or equal to 9 can be represented as the sum of three odd primes. 

6/ Another conjecture is that there are infinitely many pairs of twin primes of the form p and p + 2.