July 17, 2025

Eva Zaman 

10th Grade

Thomas A. Edison CTE High School

Prime numbers: the concept of numbers that have only two factors, one and itself. It is a lesson children are taught in elementary school, yet it has proven time and time again to be the basis for a centuries-long mystery. While its origins lie in the Greek school, Pythagoras, from 500 BC to 300 BC, and Euclid, widely considered the father of geometry, proved prime numbers to be infinite. However, this only posed another inquiry: Is there a pattern to prime distribution? This question has stumped mathematical geniuses for decades, so much so that it is one of the seven millennium questions. Created by the Clay Mathematics Institute of Cambridge, Massachusetts, in 2000, the millennium questions are a set of seven, now six, unproven series fundamental to science and mathematics. Anyone capable of solving one of the questions is awarded 1 million dollars and immortalized forever in the history of science. The seven theories are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture. The Poincaré conjecture was solved and refined by 2006; it remains the only millennium problem to be solved.

Georg Bernhard Riemann, a mathematical genius from a young age, is the creator of the Riemann hypothesis, a basis for many substantial fields, and even Einstein's theory of relativity. Many came before Riemann, who led and influenced his own understanding of geometry. Some substantial figures are Leonard Euler and Carl Friedrich Gauss, Riemann’s teacher. Euler, born in 1707, standardized modern math such as the function y = f(x). With his extensive knowledge of functions, Euler proved that the Zeta function converges or approaches a limit when s >1. By continuing to expand on his findings, he further proved that zeta functions can be expressed as infinite products of an infinite series. Through these findings, he was able to start the ties between the Zeta function and prime numbers. In 1796, Carl Friedrich Gauss made several notable contributions to mathematics. His specific interest in prime numbers presented itself when he calculated three million prime numbers to graph in what is known as the “Prime Counting Function” to represent prime numbers visually. Through the use of this graph, he was able to determine that the graph of 1/log^x was very similar to his own Prime Counting Function. Thus, Gauss’s conjecture came to be, the idea stating that the proportion of a prime number at a given point is approximately 1/log^x. 

Bernhard Riemann would later use his teacher’s discoveries to develop his very own findings. Riemann was one of the founders of complex analysis, a branch of math studying functions with inputs and outputs in a+bi form, or in other words, the use of imaginary numbers for functions. He desired to have the independent variable, s, of a Zeta function equal to a+bi, and when he did, similar to Euler’s discoveries, he found s only converged when a >1. However, unlike Euler, he used analytic continuation: the process of extending the domain of a function by creating another function that would prove the Zeta function and the newly created Riemann-Zeta function true at the same time. Through this process, the function is extended to a complex plane, an extension of the usual number system, in which imaginary numbers and complex numbers can be plotted onto a graph, creating the 2nd dimension. Riemann discovered in the complex plane that some points would cross the origin, creating Zeta zeros. While some were trivial, for example, 4 and 6, other non-trivial zeros existed between 0 and 1. This area between 0 and 1 is referred to as the critical strip. Riemann believed all non-trivial zeros would lie on one vertical line at the exact midpoint in the critical strip; the famous Riemann Hypothesis. If proven true, it confirms the idea that all harmonics, or waves, of Zeta zeros, when added up, will perfectly match Gauss’s Prime Counting Function. Therefore, it will effectively prove that the distribution of prime numbers can be predicted. 

Even 166 years after its creation, it remains unsolved, but many have tried relentlessly. To prove the hypothesis true, every single non-trivial Zeta Zero must be proven to lie on the critical line, a hard feat for a sequence of numbers that goes on for infinity. While computers have tried, not even a robot can go on forever. Nonetheless, there is no reason to lose hope, as humanity advances and more people are made aware of valuable maths such as this that must be further pursued, the chances of solving it increase.