Editing Goldbach's conjecture (section)

==Heuristic justification==
[[File:Goldbach partitions of the even integers from 4 to 28 300px.png|right|thumb|upright=1.2|Sums of two primes at the intersections of three lines]]
Statistical considerations that focus on the [[prime number theorem|probabilistic distribution of prime numbers]] present informal evidence in favour of the conjecture (in both the weak and strong forms) for [[sufficiently large]] integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.

[[File:Goldbach-1000000.png|thumb|right|upright=1.2|Number of ways to write an even number {{mvar|n}} as the sum of two primes {{OEIS|A002375}}]]

A very crude version of the [[heuristic]] probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The [[prime number theorem]] asserts that an integer {{mvar|m}} selected at random has roughly a {{math|{{sfrac|1|ln ''m''}}}} chance of being prime. Thus if {{mvar|n}} is a large even integer and {{mvar|m}} is a number between 3 and {{math|{{sfrac|''n''|2}}}}, then one might expect the probability of {{mvar|m}} and {{math|''n'' − ''m''}} simultaneously being prime to be {{math|{{sfrac|1|ln ''m'' ln(''n'' − ''m'')}}}}. If one pursues this heuristic, one might expect the total number of ways to write a large even integer {{mvar|n}} as the sum of two odd primes to be roughly

: <math>\sum_{m=3}^\frac{n}{2} \frac{1}{\ln m} \frac{1}{\ln(n - m)} \approx \frac{n}{2 (\ln n)^2}.</math>

Since {{math|ln ''n'' ≪ {{sqrt|''n''}}}}, this quantity goes to infinity as {{mvar|n}} increases, and one would expect that every large even integer has not just one representation as the sum of two primes, but in fact very many such representations.

This heuristic argument is actually somewhat inaccurate because it assumes that the events of {{mvar|m}} and {{math|''n'' − ''m''}} being prime are [[statistical independence|statistically independent]] of each other. For instance, if {{mvar|m}} is odd, then {{math|''n'' − ''m''}} is also odd, and if {{mvar|m}} is even, then {{math|''n'' − ''m''}} is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if {{mvar|n}} is divisible by 3, and {{mvar|m}} was already a prime other than 3, then {{math|''n'' − ''m''}} would also be [[coprime]] to 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully, [[G. H. Hardy]] and [[John Edensor Littlewood]] in 1923 conjectured (as part of their ''[[Twin prime#First Hardy–Littlewood conjecture|Hardy–Littlewood prime tuple conjecture]]'') that for any fixed {{math|''c'' ≥ 2}}, the number of representations of a large integer {{mvar|n}} as the sum of {{mvar|c}} primes {{math|''n'' {{=}} ''p''<sub>1</sub> + ⋯ + ''p<sub>c</sub>''}} with {{math|''p''<sub>1</sub> ≤ ⋯ ≤ ''p<sub>c</sub>''}} should be [[asymptotic analysis|asymptotically]] equal to

: <math>\left(\prod_p \frac{p \gamma_{c,p}(n)}{(p - 1)^c}\right)
 \int_{2 \leq x_1 \leq \cdots \leq x_c: x_1 + \cdots + x_c = n} \frac{dx_1 \cdots dx_{c-1}}{\ln x_1 \cdots \ln x_c},</math>

where the product is over all primes {{mvar|p}}, and {{math|''γ''<sub>''c'',''p''</sub>(''n'')}} is the number of solutions to the equation {{math|''n'' {{=}} ''q''<sub>1</sub> + ⋯ + ''q<sub>c</sub>'' mod ''p''}} in [[modular arithmetic]], subject to the [[Constraint (mathematics)|constraints]] {{math|''q''<sub>1</sub>, …, ''q<sub>c</sub>'' ≠ 0 mod ''p''}}. This formula has been rigorously proven to be asymptotically valid for {{math|''c'' ≥ 3}} from the work of [[Ivan Matveevich Vinogradov]], but is still only a conjecture when {{math|''c'' {{=}} 2}}.{{Citation needed|date=January 2016}} In the latter case, the above formula simplifies to 0 when {{mvar|n}} is odd, and to

: <math>
 2 \Pi_2 \left(\prod_{p \mid n; p \geq 3} \frac{p - 1}{p - 2}\right) \int_2^n \frac{dx}{(\ln x)^2}
 \approx 2 \Pi_2 \left(\prod_{p \mid n; p \geq 3} \frac{p - 1}{p - 2}\right) \frac{n}{(\ln n)^2}
</math>

when {{mvar|n}} is even, where {{math|Π<sub>2</sub>}} is [[Twin prime#First Hardy–Littlewood conjecture|Hardy–Littlewood's twin prime constant]]

: <math>\Pi_2 := \prod_{p\;{\rm prime} \ge 3} \left(1 - \frac{1}{(p-1)^2}\right) \approx 0.66016\,18158\,46869\,57392\,78121\,10014\dots</math>

This is sometimes known as the ''extended Goldbach conjecture''. The strong Goldbach conjecture is in fact very similar to the [[Twin prime#Twin prime conjecture|twin prime]] conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

[[File:goldbachs comet.gif|thumb|upright=1.2|Goldbach's comet; red, blue and green points correspond respectively the values 0, 1 and 2 modulo 3 of the number.]]

The ''Goldbach partition function'' is the function that associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks like a [[comet]] and is therefore called [[Goldbach's comet]].<ref>{{cite journal | last1 = Fliegel | first1 = Henry F. | last2 = Robertson | first2 = Douglas S. | year = 1989 | title = Goldbach's Comet: the numbers related to Goldbach's Conjecture | journal = Journal of Recreational Mathematics | volume = 21 | issue = 1| pages = 1–7 }}</ref>

Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as the sum of two primes, and also that the number of these representations depend strongly on the value modulo 3 of the number.