![]() ![]() Statistical considerations that focus on the 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. Heuristic justification Sums of two primes at the intersections of three lines One record from this search is that 3 325 581 707 333 960 528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 ×10 18 (and double-checked up to 4 ×10 17) as of 2013. With the advent of computers, many more values of n have been checked T. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to n = 100 000. However, the converse implication and thus the strong Goldbach conjecture remain unproven.įor small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. The weak conjecture would be a corollary of the strong conjecture: if n − 3 is a sum of two primes, then n is a sum of three primes. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the Annals of Mathematics Studies series in 2015 and has been undergoing further review and revision since. A weaker form of the second modern statement, known as " Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts thatĮvery odd integer greater than 7 can be written as the sum of three odd primes.Ī proof for the weak conjecture was proposed in 2013 by Harald Helfgott. ![]() It is also known as the " strong", "even", or "binary" Goldbach conjecture. The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. That is, the second and third modern statements are equivalent, and either implies the first modern statement. In any case, the modern statements have the same relationships with each other as the older statements did. The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer n, could not possibly rule out the existence of such a specific counterexample N). For example, if there were an even integer N = p + 1 larger than 4, for p a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). These modern versions might not be entirely equivalent to the corresponding original statements. Every even integer greater than 2 can be written as the sum of two primes. ![]()
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