## Section: New Results

### Proof and computation

Participants : Laurent Théry [correspondant] , Benjamin Grégoire.

We have been continuing our effort to improve the computing power of Coq. This has led to two "computational proof":

The Erdös conjecture for n = 2 was proved this year using a SAT solver. We succeeded to formally prove this instance in Coq independently checking the 3Gb trace of the SAT solver .

The weak Goldbach conjecture was proved last year by Harald Helfgott. This proof requires a computation that the conjecture holds for numbers less than ${10}^{28}$. This is done in two stages. The first one is to verify Goldbach conjecture for numbers less than ${10}^{18}$. The second one is to verify the weak Goldbach conjecture for numbers less than ${10}^{28}$ using a ladder with intervals ${10}^{18}$. The second stage has been completely verified in Coq. We are currently working on improving the computation power of Coq to make it possible to perform the first stage in reasonable time.