Primality, Fractality, and Image Analysis
This paper deals with the hidden structure of prime numbers. Previous numerical studies have already indicated a fractal-like behavior of prime-indexed primes. The construction of binary images enables us to generalize this result. In fact, two-integer sequences can easily be converted into a two-co...
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doaj-3bffadb6d05747a9bde88750924c093d2020-11-25T02:16:03ZengMDPI AGEntropy1099-43002019-03-0121330410.3390/e21030304e21030304Primality, Fractality, and Image AnalysisEmanuel Guariglia0Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II, 80126 Naples, ItalyThis paper deals with the hidden structure of prime numbers. Previous numerical studies have already indicated a fractal-like behavior of prime-indexed primes. The construction of binary images enables us to generalize this result. In fact, two-integer sequences can easily be converted into a two-color image. In particular, the resulting method shows that both the coprimality condition and Ramanujan primes resemble the Minkowski island and Cantor set, respectively. Furthermore, the comparison between prime-indexed primes and Ramanujan primes is introduced and discussed. Thus the Cantor set covers a relevant role in the fractal-like description of prime numbers. The results confirm the feasibility of the method based on binary images. The link between fractal sets and chaotic dynamical systems may allow the characterization of the Hénon map only in terms of prime numbers.https://www.mdpi.com/1099-4300/21/3/304binary imageCantor setHénon mapMinkowski islandprime-indexed primesRamanujan primes |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Emanuel Guariglia |
spellingShingle |
Emanuel Guariglia Primality, Fractality, and Image Analysis Entropy binary image Cantor set Hénon map Minkowski island prime-indexed primes Ramanujan primes |
author_facet |
Emanuel Guariglia |
author_sort |
Emanuel Guariglia |
title |
Primality, Fractality, and Image Analysis |
title_short |
Primality, Fractality, and Image Analysis |
title_full |
Primality, Fractality, and Image Analysis |
title_fullStr |
Primality, Fractality, and Image Analysis |
title_full_unstemmed |
Primality, Fractality, and Image Analysis |
title_sort |
primality, fractality, and image analysis |
publisher |
MDPI AG |
series |
Entropy |
issn |
1099-4300 |
publishDate |
2019-03-01 |
description |
This paper deals with the hidden structure of prime numbers. Previous numerical studies have already indicated a fractal-like behavior of prime-indexed primes. The construction of binary images enables us to generalize this result. In fact, two-integer sequences can easily be converted into a two-color image. In particular, the resulting method shows that both the coprimality condition and Ramanujan primes resemble the Minkowski island and Cantor set, respectively. Furthermore, the comparison between prime-indexed primes and Ramanujan primes is introduced and discussed. Thus the Cantor set covers a relevant role in the fractal-like description of prime numbers. The results confirm the feasibility of the method based on binary images. The link between fractal sets and chaotic dynamical systems may allow the characterization of the Hénon map only in terms of prime numbers. |
topic |
binary image Cantor set Hénon map Minkowski island prime-indexed primes Ramanujan primes |
url |
https://www.mdpi.com/1099-4300/21/3/304 |
work_keys_str_mv |
AT emanuelguariglia primalityfractalityandimageanalysis |
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