An Analysis of Geometrically Recursive Prime Numbers

Overview

This project innovates the Ulam Spiral and Klauber Triangle figures through the use of their integer orientation in pentagons. These four figures will be analyzed for geometric alignments based on most prime numbers under the integer 300, and be expressed as recursive functions to elongate the series to their first 100 terms. These rates at which prime numbers are being produced from these sequences will determine whether there are any evident patterns and if the Prime Number Theorem is supported. The pentagonal pyramid was shifted to provide the same interpretation while using a triangular grid paper. For the pentagonal spiral, a series of trigonometric functions was used to create the innermost pentagon, and the calculations for the apothem and the vector from the center of the pentagon to an intersection coordinate that isn’t on an axis helped determine the functions for the sides of the larger, concentric pentagons. Prime number alignments tested, either coming from an edge of the figure’s shape or one of the two off course sequences tested per figure, were then converted to recursive functions that could produce terms that were factored to determine the effectiveness of prime number production. The study failed to produce any significant recurring patterns between the first and second set of 50 prime numbers, largely due to the substantially decreasing rate of prime numbers production in the latter investigated terms. Amongst the 20 functions tested, only 4 of them had more than 40 of the first 100 terms being prime numbers. While 17 of the sequences had at least 4 prime numbers in their first 10 terms, this numbers decreased to 4 sequences in 41 through 50, and ended with only 2 sequences from the last 10 tested terms. The most effective function tested was an off course alignment in the Klauber Triangle, which started with 16 prime numbers before busting in the seventeenth term with the value 289, or 17 squared. This pattern of the first composite number being a squared number was only seen in 4 of the 6 functions tested from the pentagonal spiral, as well as notably recognized from the famous quadratic function discovered by Euler. While the study showed the unreliability of using recursive sequences to produce significant rates or patterns between prime numbers, the decreasing frequency was able to uphold and support the concepts stated in the Prime Number Theorem.

Competition history

• AJAS 2018 Category not listed

Related projects

Closest projects by meaning, across every fair and year in the corpus.

Browse more like this

• All Mathematics projects
• AJAS Mathematics projects
• Every AJAS 2018 project