Goldbach Conjecture

The Goldbach conjecture is a still unproved proposition that any even integer (strictly greater than 2) can be written as the sum of 2 prime numbers.

Informatically, it is verified for all the even integers up to several billions of billion.

The conjecture was proposed by the Prussian mathematician Christian Goldbach in 1742 in a letter to Leonhard Euler.

As the name suggests it is a conjecture, so to date it has no mathematical demonstration.

Mathematicians suppose it to be true, and it is computer-verified up to very large numbers, but it does not prove that it is true for all numbers.

Recent works like Vinogradov's theorem have shown that certain odd numbers can be expressed as the sum of three prime numbers. Or Terence Tao who demonstrated that any odd number can be written as the sum of at most 5 prime numbers.

Goldbach's weak conjecture is a hypothesis in number theory stating that any odd number strictly greater than 5 can be expressed as the sum of three prime numbers.

This conjecture is called weak because, if Goldbach's strong conjecture were proven, it would also lead to the validity of the weak conjecture. Indeed, if any even number greater than or equal to 6 can be decomposed into a sum of two prime numbers, adding the prime number 3 makes it possible to obtain a sum of three prime numbers, thus producing odd numbers greater than or equal to 9.

The only way to decompose 2 into a sum (called list of partitions of 2) are: 1+1 and 0+2, as 0 and 1 are not prime numbers, it is not possible to verify Goldbach's conjecture for the number 2.

The algorithm is similar to that of a prime factors decomposition. It is possible to speed up the calculations by using an already calculated list of prime numbers.

// Javascript limited to n = 200
var pr = new Array(3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97);
function goldback(n) {
for (p in pr) {
if (pr[p] <= n/2 && in_array(n-pr[p], pr)) {
return n+=+pr[p]+++(n-pr[p]);
}
}
}