[Paper Review] The distribution of prime numbers on the square root spiral

Harry K. Hahn, Robert L. Sachs|arXiv (Cornell University)|Jan 9, 2008

Graph theory and applications1 references3 citations

TL;DR

This paper investigates the distribution of prime numbers on the square root spiral (also known as the Spiral of Theodorus or Wurzelspirale), demonstrating that primes accumulate along specific spiral graphs defined by quadratic polynomials. The authors show these patterns arise from mathematical structures tied to second differences, with key examples like Euler's polynomial x² + x + 41 appearing as three distinct spiral graphs, and reveal symmetries involving square numbers and Fibonacci sequences.

ABSTRACT

Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -- second difference -- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.

Motivation & Objective

• To analyze the geometric distribution of prime numbers on the square root spiral.
• To identify the mathematical origin of spiral patterns where primes accumulate.
• To establish a connection between quadratic polynomials and the observed spiral structures.
• To explore the role of square numbers and Fibonacci sequences in shaping the spiral's symmetry.
• To compare findings with the Ulam spiral using Robert Sachs' Number-Spiral framework.

Proposed method

• Mapping natural numbers onto the square root spiral using polar coordinates with radial distance √n and angle θ = 2π√n.
• Identifying spiral graphs as curves where numbers with constant second difference lie, derived from quadratic polynomials.
• Analyzing the algebraic form of spiral graphs as y = ax² + bx + c, with a fixed second difference of 2a.
• Using the Number-Spiral model by Robert Sachs to compare patterns with the Ulam spiral and validate observations.
• Classifying spiral systems based on direction of rotation and second difference, revealing symmetry in prime and composite number placement.
• Examining divisibility by fixed prime factors and their alignment along specific spiral graphs.

Experimental results

Research questions

• RQ1 Why do prime numbers cluster along specific spiral graphs on the square root spiral?
• RQ2 What mathematical structure underlies the formation of these spiral graphs?
• RQ3 How are Euler's polynomial x² + x + 41 and other quadratic polynomials reflected in the spiral structure?
• RQ4 What is the role of square numbers and Fibonacci sequences in the symmetry of the spiral?
• RQ5 How does the square root spiral compare to the Ulam spiral in terms of prime number distribution?

Key findings

• Prime numbers accumulate along defined spiral graphs on the square root spiral, each corresponding to a quadratic polynomial with a fixed second difference.
• Euler's polynomial x² + x + 41 appears as three distinct spiral graphs on the square root spiral, each generated by a different quadratic polynomial.
• All natural numbers divisible by a given prime factor lie on specific spiral graphs, revealing a structured distribution of composites.
• Perfect square numbers (4, 9, 16, 25, 36) form a three-symmetrical system of three spiral graphs, dividing the spiral into three equal sectors.
• Fibonacci number sequences are embedded in the spiral’s structural framework, contributing to its inherent symmetry.
• The square root spiral exhibits a patterned organization of primes and composites that mirrors but differs from the Ulam spiral, with stronger geometric and algebraic coherence.