Goldbach wrote a letter to Euler in 1742 suggesting that every
integer n > 5 is the sum of three primes. Euler replied
that this is equivalent to every even n > 2 is the sum of two
primes--this is now known as Goldbach's conjecture. Schnizel
showed that Goldbach's conjecture is equivalent to every integer
n > 17 is the sum of three distinct primes.
It has been proven that every even integer is the sum
of at most six primes [Ramaré95] (Goldbach's
conjecture suggests two) and in 1966 Chen proved every sufficiently
large even integer is the sum of a prime plus a number with no more
than two prime factors (a P_{2}). In 1993 Sinisalo verified
Goldbach's conjecture for all integers less than 4^{.}10^{11}
[Sinisalo93].
More recently Jean-Marc Deshouillers, Yannick Saouter and Herman te
Riele have verified this up to 10^{14} with the help, of a Cray
C90 and various workstations. In July 1998, Joerg Richstein completed
a verification to 4^{.}10^{14} and placed a
list
of champions online. More recent work by
Oliveira e Silva has extended this to at least 4^{.}10^{17}.
See [Ribenboim95]
and [Wang84] for more information.
The Odd Goldbach Problem: Every odd n > 5 is the sum
of three primes.
There has been substantial progress on this, the easier case of Goldbach's
conjecture. In 1937 Vinogradov proved that this is true for sufficiently
large odd integers n. In 1956 Borodzkin showed n
> 3^{14348907} is sufficient (the exponent is 3^{15}).
In 1989 Chen and Wang reduced this bound to 10^{43000}.
The exponent still must be reduced dramatically before we can use computers
to take care of all the small cases.
Every even number is the difference of two primes.
Chen's work mentioned in the discussion of the Goldbach conjecture also showed that every even number is
the difference between a prime and a P_{2}.
For every even number 2n are there infinitely many pairs
of consecutive primes which differ by 2n.
Conjectured by Polignac 1849. When n=1 this is the twin prime conjecture. It is easy to show that for every positive
integer m there is an even number 2n such that there are
more than m pairs of consecutive primes with difference 2n.
In 1919 Brun proved that the sum of the reciprocals of the twin primes
converges, as so the sum B = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13)
+ (1/17 + 1/19) + ... is Brun's constant. B = 1.902160577783278...
See the Prime Glossary's entry on the twin prime conjecture.
Are there infinitely many primes of the form n^{2}+1?
There are infinitely many of the forms n^{2}+m^{2}
and n^{2}+m^{2}+1. A more general
form of this conjecture is if a, b, c are relatively prime, a is
positive, a+b and c are not both even,and b^{2}-4ac is not a
perfect square, then there are infinitely many primes an^{2}+bn+c
[HW79, p19].
The number of Fermat primes is finite.
Hardy and Wright give an argument for this conjecture in their well
known footnote [HW79,
p15] which goes roughly as follows. By the prime number theorem the probability
that a random number n is prime is at most a/log(n)
for some choice of a. So the expected number of Fermat
primes is at most the sum of a/log()
< a/2^{n}, but this sum is a.
However, as Hardy and Wright note, the Fermat numbers do not behave
"randomly" in that they are pairwise relatively prime...
Is there always a prime between n^{2} and (n+1)^{2}?
In 1882 Opperman stated pi(n^{2}+n) > pi(n^{2})
> pi(n^{2}-n) (n>1), which also seems
very likely, but remains unproven [Ribenboim95, p248].
Both of these conjectures would follow if we could prove the conjecture
that the prime gap following
a prime p is bounded by a constant times (log p)^{2}.