Goldbach's conjecture

Problem's Description

Goldbach's original conjecture, referred to as the "ternary" Goldbach conjecture, was communicated in a letter to Euler on June 7, 1742. [1]. It suggests that:

Every number greater than 2 can be expressed as the sum of three primes.

Goldbach's convention included treating 1 as a prime, a practice no longer followed. Euler rephrased the conjecture as the "strong" or "binary" Goldbach conjecture, asserting that all positive even integers greater than or equal to 4 can be written as the sum of two primes. A pair of primes $(p, q)$ such that $p + q = 2n$ for a positive integer $n$ is sometimes termed a Goldbach partition (Oliveira e Silva).

The conjecture that all odd numbers greater than or equal to 9 can be expressed as the sum of three odd primes is termed the "weak" Goldbach conjecture.

Vinogradov (1937) proved that every sufficiently large odd number is the sum of three primes and Estermann (1938) demonstrated that almost all even numbers are the sums of two primes.

Vinogradov's original "sufficiently large" $$N \geq 3^{3^{15}} \approx e^{e^{16.573}} \approx 3.25 \times 10^{6846168}$$ was subsequently reduced to $$e^{e^{11.503}} \approx 3.33 \times 10^{43000}$$ by Chen and Wang (1989).

A stronger version of the weak conjecture, known as Levy's conjecture, states that every odd number greater than or equal to 7 can be expressed as the sum of a prime and twice a prime.

An equivalent statement of the Goldbach conjecture asserts that for every positive integer $m$, there exist primes $p$ and $q$ such that $\phi(p) + \phi(q) = 2m$, where $\phi(x)$ represents Euler's totient function. Erdős and Moser contemplated relaxing the requirement that $p$ and $q$ be prime numbers in this equation as a potentially simpler means of determining if such numbers always exist.

Other variations of the Goldbach conjecture include propositions that every even number greater than or equal to 6 is the sum of two odd primes, and that every integer greater than 17 is the sum of exactly three distinct primes.

Let $R(n)$ denote the number of representations of an even number $n$ as the sum of two primes.

The "extended" Goldbach conjecture posits that the number of representations of an even number $n$ as the sum of two primes is given by

$$ \sim 2 \Pi_2 \prod_{k=2; p_k | n} \frac{p_k - 1}{p_k - 2} \int_2^n \frac{dx}{(\ln x)^2}, $$

where $$\Pi_2 := \prod_{p \text{ prime} \geq 3} \left(1 - \frac{1}{(p-1)^2}\right) \approx 0.66016\,18158\,46869\,57392\,78121\,10014\ldots $$ is Hardy–Littlewood's twin primes constant.

For other related problems see

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture#cite_note-2

1. OtherIs an originChristian Goldbach's letter to Leonhard Euler (letter XLIII)

   • Christian Goldbach

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