Carlos Correa

This simple work proposes an elementary solution to the conjecture: "There are infinitely many primes of the form n^2+1", mentioned by German mathematician Edmund Landau at the 5th International Congress of Mathematics held in Cambridge in 1912.

In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which allows us to limit the error with which said theorem predicts the value of the Primecounting function π(x); and, in this way, endorse the veracity of the Riemann Hypothesis.

En este trabajo se propone una solución muy simple a la conjetura de Goldbach, partiendo de una versión modificada de la criba de Eratóstenes. Introducción En 1912, en el V Congreso de Matemática realizado en Cambridge, el matemático alemán Edmund Landau planteó, entre otras, una conjetura que puede enunciarse usando conocimientos matemáticos básicos, pero que sin embargo permanece sin resolver desde hace siglos. Se trata de la conjetura de Goldbach. La conjetura de Goldbach Planteada en 1742, a través de una carta enviada por Christian Goldbach a Leonhard Euler, La conjetura dice que: "Todo número par mayor que 2 puedes escribirse como la suma de dos números primos".

This work involves the use of computational tools, capable to analyze large quantities of data, to give credibility, by searching for counterexamples, to a new conjecture about coprime numbers. This conjecture states that: "Between two pairs of consecutive multiples of all natural number k>1 always there is at least one number that is relative prime of all natural numbers ≤ k".   
The more the counterexamples search is extended, the stronger the raised conjecture will be. Therefore, in order to make this search to be also efficient, techniques of parallel high-performance SIMD computing are explored and applied in this research work. Specifically, it is the use of GPU Computing and CUDA programming as a platform for the implementation and testing of sieve algorithms wich are specially designed for finding coprimes. One of these algorithms was designed strictly observing the conjecture, but it is the one with higher demand for computational resources (time & memory). The second algorithm was designed adding restrictions to the original conjecture. This idea allowed, at the risk of finding pseudo-counterexamples, the reduction of computational resources demand. Anyway, there always exists the possibility of checking the original conjecture on a point of the search in which any of these pseudo-counterexamples could appear.
Keywords: Number Theory, Parallel Processing, GPU Computing, CUDA, C++.

It is a program, written in C ++ language, that allows to find the prime numbers, and pi (n), within any interval [a-b] of up to 16.000.000 of numbers wide.The results (primes> 2) are stored in a text file called "PrimesAtoB" and accumulate, progressively, with each program run.

En este trabajo se propone y demuestra una nueva conjetura relativa a la distancia o gap máximo entre números coprimos. Luego, probada la veracidad de dicha conjetura, se demuestra la veracidad de las siguientes conjeturas no resueltas aún:
- Conjetura de Andrica
- Conjetura de Legendre
- Distancia máxima entre primos consecutivos

This work proposes and proves a new conjecture on the maximum gap existing between coprime numbers. Then, upon proving the veracity of said conjecture, the following still-unsolved conjectures are proven true as well:

- Andrica’s Conjecture
- Legendre’s Conjecture
        - Maximum prime gap