Goldbachâ€™s conjecture states that every even integer > 2 is the sum of 2 primes. An example is 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. An alternative statement of the conjecture is as follows. Every integer > 3 is the arithmetic mean of 2 primes. This means that every integer > 3 is midway between 2 primes. For instance, 9 is midway between 5 and 13. 9 is also midway between 7 and 11.

Imagine a finite part of the number line up to some integer, N. Let this number line be divided up into roughly 4 equal parts. If it can be shown that every integer in the 2nd quarter of this number line is the arithmetic mean of 2 primes, then Goldbachâ€™s conjecture is proved. So prime number 3 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. The next prime number, 5 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. Some of these arithmetic means will be repetitions of those produced using prime number 3. The number of new arithmetic means produced by prime number 5 in the 2nd quarter of the number line is equal to the number of arithmetic means produced by prime number 5 minus the number of twin primes (5 â€“ 3 = 2) mainly in the 3rd and 4th quarters of the number line. The next prime number, 7 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. Some of these arithmetic means will be repetitions of those produced using prime number 3 and prime number 5. The number of new arithmetic means produced by prime number 7 in the 2nd quarter of the number line is equal to the number of arithmetic means produced by prime number 7 minus the number of twin primes (5 â€“ 3 = 2), mainly in the 3rd and 4th quarters of the number line, minus the number of primes that differ by 4 (7 â€“ 3 = 4), mainly in the 3rd and 4th quarters of the number line. This process carries on until all arithmetic means in the 2nd quarter are produced and any further process will produce only repetitions of arithmetic means. The phrase, â€˜mainly in the 3rd and 4th quarters of the number lineâ€™, is used because primes in the second quarter will also be involved in producing arithmetic means in the second quarter. The mathematical justification of the last but one sentence will now be discussed.

The number of primes less than N is approximately equal to N / log e N. This is a lower bound for N â‰¥ 17. An upper bound for the number of prime difference, h (not necessarily consecutive primes) < N is 8N / (log e N)^2. For instance, for the twin primes, h = 2 and therefore the number of twin primes < N cannot exceed 8N / (log e N)^2. This can be shown using sieve theory. It should be clear from the discussion so far that for each cycle of production of arithmetic means, the primes involved in the 2nd, 3rd and 4th quarters are such that the difference between the largest and smallest prime is approximately N/2. So, prime number 3 uses approximately N / log e N â€“ [N/2]/[ log e (N/2)] primes to produce the same number of arithmetic means in the 2nd quarter of the number line. For large enough N, N / log e N â€“ [N/2]/[ log e (N/2)] can be approximated to [N/2]/[ log e (N/2)] since log e 2 = 0.693, which is small compared to log e N. For large enough N, approximately [N/2]/[ log e (N/2)] primes in the 2nd, 3rd and 4th quarters will be used for each cycle. Let c be the number of cycles required to produce all the N/4 arithmetic means in the 2nd quarter. The following equation can now be formed using the upper bound 8N / (log e N)^2.

c[N/2]/[ log e (N/2)] â€“ 4cN / (log e N)^2 = N/4

The approximate number of cycles, c, required using an upper bound of 8N / (log e N)^2 is

c â‰ˆ [ log e (N/2)]/2

So the number of cycles required (which is the same as the number of primes required in the first quarter) to produce all N/4 arithmetic means is much less than the number of primes in the first quarter i.e. c â‰ˆ [ log e (N/2)]/2 < [N/4]/[ log e (N/4)]. This proves that Goldbachâ€™s conjecture is true.