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Goldbach's conjecture – and more:

Even numbers n > 4208 are sums of two primes (twins!)

*Prime twins* (or *twin primes*)
are odd prime numbers that come in pairs: {3, 5}, {11, 13}, {17, 19}, {29, 31}.
Primes in each twin pair differ by two.
One can prove that each prime twin pair except {3, 5}
has the form {6*i* – 1, 6*i* + 1} for a certain integer *i*.
On this page, we will investigate the following hypotheses:

(A) *Every even number n* > 4208 *can be written as a sum of two twin primes.*

(B) *Every positive even number can be written as a difference of two twin primes.*

For comparison, here are two famous (weaker) conjectures:

**Goldbach's conjecture:** *Every even number n* > 2 *can be written as a sum of two primes.*

**Twin prime conjecture:** *There are infinitely many twin primes.*

If statements (A) and (B) are true, then Goldbach's conjecture must be true and the twin prime conjecture must be true, too.
The table below presents a partial computational check of the above statements (A) and (B).
For each *n*, the table shows only one of many existing representations of *n* as a difference of twin primes;
*n* = 4208 is the last line in the table showing a *difference only* (since no sums were found).
Apparently, for all even *n* ≥ 4210, representations of *n* as a *sum of two twin primes* do exist.
Note that this is *not a proof*, just an illustration: both statements (A) and (B) are plausible.
(The twin prime conjecture and Goldbach's conjecture are almost certain to be true – but not proven as of 2011.)

n # of sums -p+q p+q (n as a sum/difference of two twin primes p and q)