## Wednesday, February 9, 2011

### Goldbach conjecture

Every even integer greater than 2 can be expressed as the sum of two primes.

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### New theories reveal the nature of numbers

Online tool to test Goldbach's conjecture on submitted integers.

For example,
0 ..................... 10 .......................... 20

10 = 3 + 7 ... ...10 = 5 + 5
20 = 3 + 17 .... 20 = 7 + 13

0 ....................... 50 ......................... 100

50 = 3 + 47 ... 50 = 7 + 43 ... 50 = 13 + 37 ... 50 = 19 + 31
100 = 3 + 97 ... 100 = 11 + 89 ... 100 = 17 + 83 ... 100 = 29 + 71 ... 100 = 41 + 59 ...
100 = 47 + 53

0 ........................ n .......................... 2n

Goldbach conjecture verification

Let n be an even number larger than two, and let n=p+q, with p and q prime numbers, p<=q,
be a Goldbach partition of n. Let r(n) be the number of Goldbach partitions of n. The number of ways of writing n as a sum of two prime numbers, when the order of the two primes is important, is thus R(n)=2r(n) when n/2 is not a prime and is R(n)=2r(n)-1 when n/2 is a prime. The Goldbach conjecture states that r(n)>0, or, equivalently, that R(n)>0, for every even n larger than two.

A lesser known Goldbach Conjecture

What is known :

(1) Schnirelmann(1930): There is some N such that every number from some point onwards
can be written as the sum of at most N primes.

(2) Vinogradov(1937): Every odd number from some point onwards can be written as
the sum of 3 primes.

(3) Chen(1966): Every sufficiently large even integer is the sum of a prime and
an "almost prime" (a number with at most 2 prime factors).