# Prime Conjectures and Open Questions

### By Chris Caldwell

Below are just a few of the many conjectures concerning primes.

**Goldbach's Conjecture: Every even***n*> 2 is the sum of two primes.- 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 2***n*are there infinitely many pairs of*consecutive*primes which differ by 2*n*.- 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 2*n*such that there are more than*m*pairs of consecutive primes with difference 2*n*. **Twin Prime Conjecture: There are infinitely many twin primes.**- 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*[HW79, p19].^{2}-4ac is not a perfect square, then there are infinitely many primes an^{2}+bn+c **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}.

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