Purpose

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.

Design/methodology/approach

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.

Findings

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.

Originality/value

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.

中文翻译：

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乌拉姆螺旋中相邻素数的分布

目的

自然数在乌拉姆螺旋中的分布体现了难以捉摸的素数的一系列意想不到的规律。他们的排序仍然是一个研究兴趣的主题，在数学的不同分支中有许多分支，特别是在数论和素因数分解问题中。因此，研究的重点是最著名和最广泛的非对称密码系统，即 RSA 加密。

设计/方法/方法

考虑到 Hardy-Littlewood 孪生素数猜想和素数星座，本文介绍了出现在螺旋中的不同素数的 1、2、3 或 4 个邻接关系。

发现

通过方程，给出了相邻位置有1到4个素数的螺旋内素数的计算公式，并附有便于运算的近似表达式，表明随着螺旋的进行，邻接度下降的非常快，但并没有消失。

原创性/价值

提供了对素值的距离根据 Ulam 螺旋的步长以上升方式变化的情况的概括。