In our case we use the binary band to differentiate primes from non primes.
#LIST OF PRIME NUMBERS IN BINARY SERIES#
These bands are a series of boxes painted in different colors, where each color represents any specific property of the number. In order to visualize these classifications we use the binary bands.
Here we present this classification and their corresponding formulas:Įxcept 2 and 3, all the others prime numbers are of the form as 1+6.n and 5+6.n. The classification in six families is the lowest possible and determines that all primes greater than 3 belong to two different families. These patterns can be observed in the following pictures at Figures 1 and 2.įigure 1: First prime patterns obtained by the use of digital rootįigure 2: Sequential zoom of the pattern and the hole figure up to 1000Īccording to this scheme we can classify all numbers in nine families one for each column and even in only six families. As expected we find patterns of primes, but the most remarkable thing is that these patterns are reflected vertically and mirrored horizontally when we use negative primes. In order to follow the location of primes we painted the boxes which contain prime numbers. By doing this we can see that prime numbers are located in certain columns. It is evident that these groups of primes, or constellations, are amazing because the whole prime structure has no evidence of certain order, it has no series expansion, and has no symmetry.Ģ ABOUT PRIME LOCATIONS AND THE USE OF DIGITAL ROOTīased on a previous work we will use the digital root in order to establish a matrix representation of numbers in nine columns, each of them corresponding to a determined digital root. Recently, it has been demonstrated that there exist arbitrary large arithmetic progressions of primes (Terence Tao). For that reason it is remarkable the existence of twin primes, (prime numbers separated by 2), Cousin primes (primes separated by 4), sexy primes (primes separated by 6), and even triplets and cuadruplets of primes. The location of prime numbers does not follow a defined pattern, and it has an evident lack of symmetry. Nowadays, prime search algorithms are based on probabilistic mathematic structures, although some mathematicians think that prime structure is totally determined by certain rules. Nevertheless, no one has found a single formula that gives all prime numbers as a result. Some of them found formulas that based on natural numbers will give possible primes, like Fermat and Merssene. Finally we will see that some unsolved problems like Goldbach conjecture and related ones, are based in hidden symmetries to be proved.įor centuries many mathematicians, scientists and erudits studied the problem of the location of prime numbers. By means of this method we will see that the position of prime numbers is far away from being random, on the contrary, it has a totally determined structure, at least not directly.
Those layers have a totally determined frequency and they are fully symmetric. Copying Eratosthenes we introduce a sieve in the form of layers of prime multiples represented as binary bands.
Gustavo Funes, Damian Gulich, Leopoldo Garavaglia and Mario GaravagliaĪddress: Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, y Laboratorio de Procesamiento Láser, Centro de investigaciones Ópticas, La Plata, Argentina.Īddress: Calle del General Pingarrón Nº 3Įmail: We have established a new way of studying simple mathematical problems in a graphical way.