Tuesday, November 2, 2021

What is the Goldbach’s conjecture?

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 whole number greater than 2 is the sum of two prime numbers.

There is a Goldbach´s weak conjecture (every odd integer greater than 5 can be written as the sum of three odd primes) and strong Goldbach conjecture (every even integer greater than 2 can be written as the sum of two primes).

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov’s method, Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann showed that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1).

For small values of n (positive even integer), the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to n ≤ 105.

With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 1018 (and double-checked up to 4 × 1017) as of 2013.

One record from this search is that 3325581707333960528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.

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:
  • It was proven by Lagrange 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 (n > 5) is the sum of a prime and the double of a prime.”
  • 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.
  • A strengthening of the Goldbach conjecture proposed by Harvey Dubner states that every even integer greater than 4,208 is the sum of two twin primes. Only 33 even integers less than 4,208 are not the sum of two twin primes.

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