Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

The conjecture has been shown to hold for all natural numbers less than , but remains unproven despite considerable effort.

History

Origins

On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture:

Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first:{\ln x_1 \cdots \ln x_c},</math>

where the product is over all primes , and is the number of solutions to the equation in modular arithmetic, subject to the constraints . This formula has been rigorously proven to be asymptotically valid for from the work of Ivan Matveevich Vinogradov, but is still only a conjecture when . In the latter case, the above formula simplifies to 0 when is odd, and to

<math display="block">

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 is even, where is Hardy–Littlewood's twin prime constant

<math display="block">\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 Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

Goldbach partition function

thumb|upright=1.2|class=skin-invert-image|Goldbach's comet; red, blue and green points correspond respectively the values 0, 1 and 2 modulo 3 of the number.

The function 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.

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 depends strongly on the value modulo 3 of the number.

Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations.

Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares:

  • Lagrange proved that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes.
  • Hardy and Littlewood listed as their Conjecture I: "Every large odd number () is the sum of a prime and the double of a prime". This conjecture is known as Lemoine's conjecture and is also called Levy's conjecture.
  • The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers.
  • Harvey Dubner proposed a strengthening of the Goldbach conjecture: that every even integer greater than 4208 is the sum of two twin primes (not necessarily belonging to the same pair). Only 34 even integers less than 4208 are not the sum of two twin primes; Dubner has verified computationally that this list is complete up to <math>2\cdot 10^{10}.</math> A proof of this stronger conjecture would imply not only Goldbach's conjecture but also the twin prime conjecture.

Goldbach's conjecture is used to study computational complexity. The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 27-state Turing machine that halts if and only if Goldbach's conjecture is false.

Goldbach's conjecture is part of the plot of the 2007 Spanish film Fermat's Room.

Goldbach's conjecture is featured as the main topic of research of the eponymous character Marguerite in the 2023 French-Swiss film Marguerite's Theorem.

Goldbach's conjecture is a plot device in the Frederik Pohl novella, "The Gold at the Starbow's End."

Notes

References

Further reading

  • Terence Tao proved that all odd numbers are at most the sum of five primes .
  • Goldbach Conjecture at MathWorld.
  • Goldbach's original letter to Euler &mdash; PDF format (in German and Latin)
  • Goldbach's conjecture , part of Chris Caldwell's Prime Pages.
  • Goldbach conjecture verification, Tomás Oliveira e Silva's distributed computer search.