The distribution of natural numbers in the Ulam spiral manifests a series of unexpected regularities of the elusive prime numbers. Their sequencing remains a topic of research interest, with many ramifications in different branches of Mathematics, especially in number theory and the prime factorisation problem. Accordingly, the focus of the research is on the most known and widespread asymmetric cryptographic system, that is, the RSA encryption.
This paper presents the presence of one, two, three or four adjacencies for the diverse primes that appear in a spiral, considering the Hardy–Littlewood twin prime conjecture and the constellations of primes.
Through equations, the calculation formulas of primes inside a spiral that have one to four primes in their adjacent places is offered, with approximate expressions that facilitate the operations, showing that the adjacencies decrease very rapidly as the spiral progresses, although without disappearing.
A generalisation to cases in which the distances to the prime values change in an ascending way in accordance with the step of the Ulam spiral is offered.
This study has received support from the Instituto Nacional de Ciberseguridad (INCIBE), from Ministerio de Economía y Empresa of Spain, within the framework of “Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad” (ref INCIBEI-2015–27342).
This study has received support from the National Institute of Cybersecurity (INCIBE), from the Ministry of Economy and Business of Spain, within the framework of “Helps for excellence of advanced research teams in cybersecurity” (ref INCIBEI-2015–27342).
Contributions: conceptualisation, Vicente Jara Vera; funding acquisition, Carmen Sánchez Ávila; methodology, Vicente Jara Vera; investigation, Vicente Jara Vera; supervision, Vicente Jara Vera, Carmen Sánchez Ávila; validation, Vicente Jara Vera, Carmen Sánchez Ávila; wrote the paper, Vicente Jara Vera.
Jara-Vera, V. and Sánchez-Ávila, C. (2021), "Distribution of adjacent prime numbers in the Ulam spiral", Engineering Computations, Vol. 38 No. 4, pp. 1633-1651. https://doi.org/10.1108/EC-09-2019-0402
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