"Existen infinitos primos de la forma x^2+1", 2024
En este sencillo trabajo se propone una solución elemental a la conjetura "Existen infinitos prim... more En este sencillo trabajo se propone una solución elemental a la conjetura "Existen infinitos primos de la forma x^2+1", mencionada, por el matemático alemán Edmund Landau, en el V Congreso de Matemática realizado en Cambridge.
"There are infinitely many primes of the form n^2+1", 2024
This simple work proposes an elementary solution to the conjecture: "There are infinitely many pr... more 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.
The Riemann's Hypothesis, the Prime Numbers Theorem (PNT), and the Error, 2024
In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which all... more 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.
Una solución sencilla a la cojetura de Goldbach, 2023
En este trabajo se propone una solución muy simple a la conjetura de Goldbach, partiendo de una v... more 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, t... more 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), wit... more 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.
In this work a very simple solution to the Goldbach's conjecture is proposed, starting from a mod... more In this work a very simple solution to the Goldbach's conjecture is proposed, starting from a modified version of the Sieve of Eratosthenes.
In this work, three methods for proving the existence of infinite twin prime numbers are presente... more In this work, three methods for proving the existence of infinite twin prime numbers are presented. In addition, a non-heuristic approach to counting the prime numbers § n is presented as well.
En este trabajo se proponen tres formas de avalar la existencia de infinitos primos gemelos. Adem... more En este trabajo se proponen tres formas de avalar la existencia de infinitos primos gemelos. Además, se ofrece una aproximación no heurística al conteo de primos.
En este trabajo se propone y demuestra una nueva conjetura relativa a la distancia o gap máximo e... more 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 number... more 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
"Existen infinitos primos de la forma x^2+1", 2024
En este sencillo trabajo se propone una solución elemental a la conjetura "Existen infinitos prim... more En este sencillo trabajo se propone una solución elemental a la conjetura "Existen infinitos primos de la forma x^2+1", mencionada, por el matemático alemán Edmund Landau, en el V Congreso de Matemática realizado en Cambridge.
"There are infinitely many primes of the form n^2+1", 2024
This simple work proposes an elementary solution to the conjecture: "There are infinitely many pr... more 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.
The Riemann's Hypothesis, the Prime Numbers Theorem (PNT), and the Error, 2024
In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which all... more 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.
Una solución sencilla a la cojetura de Goldbach, 2023
En este trabajo se propone una solución muy simple a la conjetura de Goldbach, partiendo de una v... more 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, t... more 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), wit... more 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.
In this work a very simple solution to the Goldbach's conjecture is proposed, starting from a mod... more In this work a very simple solution to the Goldbach's conjecture is proposed, starting from a modified version of the Sieve of Eratosthenes.
In this work, three methods for proving the existence of infinite twin prime numbers are presente... more In this work, three methods for proving the existence of infinite twin prime numbers are presented. In addition, a non-heuristic approach to counting the prime numbers § n is presented as well.
En este trabajo se proponen tres formas de avalar la existencia de infinitos primos gemelos. Adem... more En este trabajo se proponen tres formas de avalar la existencia de infinitos primos gemelos. Además, se ofrece una aproximación no heurística al conteo de primos.
En este trabajo se propone y demuestra una nueva conjetura relativa a la distancia o gap máximo e... more 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 number... more 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
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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++.
- Conjetura de Andrica
- Conjetura de Legendre
- Distancia máxima entre primos consecutivos
- Andrica’s Conjecture
- Legendre’s Conjecture
- Maximum prime gap
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++.
- Conjetura de Andrica
- Conjetura de Legendre
- Distancia máxima entre primos consecutivos
- Andrica’s Conjecture
- Legendre’s Conjecture
- Maximum prime gap