Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations

The paragraph begins with a discussion on a pattern observed in polar coordinates related to prime numbers and pi, as seen on Math Stack Exchange. It introduces the concept of polar coordinates, contrasting it with the usual Cartesian coordinates, and explains how points in 2D space are labeled with a radius and an angle (theta) measured in radians. The pattern involves plotting points where both coordinates are prime numbers, which leads to a visualization of spirals and rays when zoomed out. The paragraph also touches on the distribution of primes and the transition from chaotic spirals to orderly rays at larger scales, hinting at deeper mathematical principles.

This paragraph delves into the explanation of the spiral patterns observed in the previous section. It describes how counting by 6 leads to almost full turns, creating the illusion of a single curving line. The paragraph clarifies that prime numbers cannot be multiples of 6, which explains the absence of certain spiral arms when focusing on primes. It introduces the concept of residue classes mod 6 and explains how these sequences relate to the observed spirals. The paragraph also sets up the groundwork for understanding larger patterns by discussing the relationship between the number of steps (44 in this case) and the close approximation to full turns, leading to the concept of residue classes mod 44.

The paragraph continues the exploration of prime numbers in polar coordinates, focusing on the larger scale and the appearance of Milky Way-like gaps. It explains how the primes are distributed among the residue classes mod 44 and how this leads to the observed pattern of gaps and clumps. The paragraph introduces the concept of relatively prime numbers, also known as co-prime, and Euler's totient function, which counts the number of co-prime integers up to a given number. It also discusses the significance of 710 radians being a close approximation to a full turn and how this leads to the formation of nearly straight lines in the visualization.

This paragraph discusses the distribution of prime numbers among residue classes mod 10 and mod 44, using histograms to illustrate the even spread of primes across these classes. It challenges the expectation of an even distribution and highlights the non-obvious nature of proving such a pattern. The paragraph introduces Dirichlet's theorem, which states that the primes are equally dense in any residue class that is co-prime to the divisor, and explains its significance in number theory. It also touches on the complexity of proving this theorem and its reliance on complex analysis, emphasizing the relevance of understanding prime distribution in modern mathematical research.

The final paragraph reflects on the value of exploring mathematics through data visualization and serendipitous discovery. It emphasizes that even arbitrary explorations can lead to important and deep mathematical facts, such as Dirichlet's theorem. The paragraph argues that this kind of exploration can make mathematical concepts more familiar and easier to understand when formally introduced. It concludes by encouraging the pursuit of mathematical curiosity and the joy of rediscovery, which can lead to a more effective learning experience.