Goldbach Conjecture

The Goldbach Conjecture is a yet unproven conjecture stating that every even integer greater than two is the sum of two prime numbers. The conjecture has been tested up to 400,000,000,000,000.

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics.

For example,

   4 = 2 + 2
   6 = 3 + 3
   8 = 3 + 5
  10 = 3 + 7 = 5 + 5
  12 = 5 + 7
  14 = 3 + 11 = 7 + 7
  etc.

Origins

In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture:

   Every integer greater than 2 can be written as the sum of three primes.

He considered 1 to be a prime number, a convention subsequently abandoned. So today, Goldbach's original conjecture would be written:

   Every integer greater than 5 can be written as the sum of three primes.

Euler, becoming interested in the problem, answered with an equivalent version of the conjecture:

   Every even number greater than 2 can be written as the sum of two primes,

adding that he regarded this a fully certain theorem ("ein ganz gewisses Theorema"), in spite of his being unable to prove it.

The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" or "binary" Goldbach conjecture. The conjecture that all odd integers greater than 9 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one.

The majority of mathematicians believe the conjecture (in both the weak and strong forms) to be true, at least for sufficiently large integers, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers: the bigger the number, 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.

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 $m$ selected at random has roughly a $\frac{1}{\ln m}$ chance of being prime. Thus if $n$ is a large even integer and $m$ is a number between 3 and $\frac{n}2$ , then one might expect the probability of $m$ and $n-m$ simultaneously being prime to be $\frac{1}{\ln m \cdot \ln (n-m)}$ . This heuristic is non-rigorous for a number of reasons: for instance, it assumes that the events that $m$ and $n-m$ are prime are statistically independent of each other. Nevertheless, if one pursues this heuristic, one might expect the total number of ways to write a large even integer $n$ as the sum of two odd primes to be roughly

$\sum_{m=3}^{n/2} \frac{1}{\ln m} {1 \over \ln (n-m)} \approx \frac{n}{2 \ln^2 n}$

Since this quantity goes to infinity as $n$ increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

The above heuristic argument is actually somewhat inaccurate, because it ignores some correlations between the likelihood of $m$ and $n-m$ being prime. For instance, if $m$ is odd then $n-m$ is also odd, and if $m$ is even, then $n-m$ is even, a non-trivial relation because (besides 2) only odd numbers can be prime. Similarly, if $n$ is divisible by 3, and $m$ was already a prime distinct from 3, then $n-m$ 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, Hardy and Littlewood in 1923 conjectured (as part of their famous Hardy-Littlewood prime tuple conjecture) that for any fixed $c \geq 2$ , the number of representations of a large integer $n$ as the sum of $c$ primes $n=p_1+\dotsb+p_c$ with $p_1 \leq \dotsb \leq p_c$ should be asymptotically equal to

$\left(\prod_p \frac{p \gamma_{c,p}(n)}{(p-1)^c}\right) \int_{2 \leq x_1 \leq \dotsb \leq x_c: x_1+\ldots+x_c = n} \frac{dx_1 \ldots dx_{c-1}}{\ln x_1 \ldots \ln x_c}$

where the product is over all primes $p$ , and $\gamma_{c,p}(n)$ is the number of solutions to the equation $n = q_1 + \ldots + q_c \pmod p$ in modular arithmetic, subject to the constraints $q_1,\ldots,q_c \neq 0 \pmod p$ . This formula has been rigorously proven to be asymptotically valid for $c\geq 3$ from the work of Vinogradov, but is still only a conjecture when $c = 2$ . In the latter case, the above formula simplifies to 0 when $n$ is odd and to

$2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \int_2^n \frac{dx}{\ln^2 x} \approx 2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \frac{n}{\ln^2 n}$

when $n$ is even, where $\Pi_2$ is the twin primes contant,

$\Pi_2 := \prod_{p \geq 3} \left(1 - \frac{1}{(p-1)^2}\right) = 0.660161858\ldots$

This asymptotic is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the Twin Prime Conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

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Goldbach Conjecture

Origins