The Goldbach Conjecture

Ricardo Guillermo Barca (1)

1 UTN-FRBA - Universidad Tecnológica Nacional [Buenos Aires]

The binary Goldbach conjecture asserts that every even integer greater than 4 is the sum of two primes. In order to prove this statement, we start by defining a kind of double sieve of Eratosthenes as follows. Given a positive even integer x, we sift out from [1, x] all those elements that are congruents to 0 modulo p, or congruents to x modulo p, where p is a prime less than sqrt{x}. So, any integer in the interval [sqrt{x}, x] that remains unsifted is a prime p for which either x-p = 1 or x-p is also a prime. Then, we introduce a new way to formulate this sieve, which we call the sequence of k-tuples of remainders. Using this tool, we obtain a lower bound for the number of elements in [1, x] that survives the sifting process. We prove, for every even number x > p_{35}^2, that there exist at least 3 integers in the interval [ 1, x ] that remains unsifted. This proves the binary Goldbach conjecture for every even number x > p_{35}^2, which is our main result.

Mots clés: Sieve theory; Goldbach problem

Domaines: Théorie des nombres [math.NT]

Ricardo Guillermo Barca. The Goldbach Conjecture. 2018. ⟨hal-01662191v2⟩