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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

100 = 47 + 53

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

Goldbach conjecture verification

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.

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

(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

(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).

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