Goldbach's conjecture
The Goldbach Conjecture, named after the mathematician Christian Goldbach, is an unproven statement from the field of number theory. It belongs as one of Hilbert's problems to the most famous unsolved problems in mathematics.
Strong (or binary ) Goldbach's Conjecture
The strong (or binary ) Goldbach's conjecture is as follows:
With this assumption dealt up to the present time many number theorists, without prove it or to disprove.
Tomás Oliveira e Silva showed by means of a distributed - computing project now (April 2012) the validity of the conjecture for all numbers up to 4.1018. A proof that they for any arbitrarily large even number is true, this is not natural.
After the British publisher Faber & Faber was awarded a prize of one million dollars on the proof of the conjecture in 2000, and the public interest grew in this question. The prize money has not been paid since April 2002 no evidence was received.
Weak ( or ternary ) Goldbach's Conjecture
The weaker conjecture
Is known as ternary or weak Goldbach's Conjecture. It is partially solved: On the one hand it is true if the generalized Riemann hypothesis is true, and on the other hand it is shown that it is valid for sufficiently large numbers ( set of Vinogradov, see Related results ).
On May 13, 2013, the mathematician Harald Helfgott peruanischstämmige announced an alleged proof of the ternary Goldbach 's conjecture for all numbers. The validity for all numbers below has already been checked with computer assistance.
From the strong Goldbach 's conjecture follows the weak Goldbach's conjecture, for every odd number can be written as a sum. The first term can be written after the strong Goldbach 's conjecture as the sum of two prime numbers (a and b ), thus a decomposition of three prime numbers (a, b and 3) is found.
Goldbach decompositions
As Goldbach decomposition, the representation of an even number is called the sum of two prime numbers, such as 3 5 is a Goldbach decomposition of 8 decompositions are not unique, as one of 18 = 7 11 = 5 can 13 seen. For larger even numbers, there is a tendency growing number of Goldbach partitions ( " multiple Goldbach numbers "). The number of Goldbach decompositions can be easily calculated with the aid of computers, see figure.
In order to violate the strong Goldbach's Conjecture, one data point would eventually fall to the zero line.
Requiring an even number, that for every prime number with a prime number and thus a Goldbach decomposition is (the number that is the maximum number of Goldbach decompositions has ) satisfy exactly the four numbers 10, 16, 36 and 210 also the weaker requirement that for every prime number also is a prime number, met not a number.