Pierre de Fermat Primes have a tendency to arrange themselves in pairs of the form ( p, p+2): for example 3 and 5, 5 and 7, 17 and 19. This is also evident among much larger numbers such as 29,879 and 29,881. Such primes are called twin primes or prime pairs, and it is not known whether there are infinitely many of these twin primes. However most mathematicians believe the answer is ‘yes’.
A more famous conjecture regarding primes is the Goldbach Conjecture, named after Christian Goldbach
(1) Every even number greater than or equal to 4 is the sum of two primes; for example
(It is easy to verify that
this conjecture fails for odd numbers, 11 or example.)
In the letter Goldbach also expressed the following belief:
(2) Every integer n greater than or equal to 5 is the sum of three primes.
As far as is known, Euler did not prove (1), but neither Euler nor anyone else has been able to find a counter-example. This conjecture has since been tested for all even numbers up to at least 1010 and found to be true. This still remains one of the great unsolved conjectures of mathematics.
Pierre de Fermat
is prime for any non-negative integer n. The conjecture was proven to be incorrect by Euler in 1732 who showed that
641.More recently analysis of these
so-called Fermat numbers have found no other primes above F4. However no-one has yet proved that F4 is the largest Fermat prime.