Appalachian State University: A Brief History Of Prime Numbers

History of Prime Numbers
Brenden Rawls
Appalachian State University A prime number is a number greater than one that can only be divided by itself and one. Mathematicians consider these numbers to be the “atoms of arithmetic or a mathematician’s periodic table” (du Sautoy, 2003, p. 19). These numbers are considered to be the building blocks for number theory, and after over two millennia, we still know so little about them. Throughout history, mathematicians have asked questions about prime numbers some of which have been successfully answered and others that continue to be researched. These questions include the following: What is the largest known prime? Is there a way to predict how when the next prime will appear on the number line? How

in Alexandria, Egypt, is one of the most influential mathematicians of ancient times. He traveled the Mediterranean to further his understanding of mathematics and to study the works of earlier mathematicians. Euclid eventually settled down in Alexandria where he opened a school and taught students interested in the subjects he had mastered. Much mathematical advancement is associated with Euclid, but one stands above the rest, The Elements. A series beginning with twenty-three definitions, five axioms, and five postulates; The Elements allowed Euclid to progress step by step building his first proof. The next step was to use his assumptions and his first proof to build the proof for his second theorem. He did this for nearly 450 propositions. The idea of starting with base assumptions and building proofs off of their predecessors is now called an “Axiomatic System.” Two of Euclid’s proofs directly correlated to prime numbers: “The Prime Divisor” and “The Infinitude of Primes.” With the Prime Divisor, Euclid proved that if a number is composite then it is the product of prime numbers. Now, Euclid began to think, as primes appear on the number line they to thin out the further and further you go—he questioned if they disappeared? For example, there are twenty-five primes between zero and one hundred, but there are six between one million and one million one hundred. He started with an assumption that there existed a finite amount of primes at his

He is one of the most influential mathematicians in terms of the study of primes. Recall Euler’s Zeta function, ∑_(n=1)^∞▒1/n^s . It would be Riemann, who was a student of Gauss, to allow s to be complex numbers and further Euler’s fuction. These complex numbers were in the form of a real number + some number(i). This allowed Riemann to begin to search the zeta function and find it’s zeros in a plane with the number line as one axis and an imaginary line as another. These zeros were originally thought to be scattered throughout the number plane, “Euler had made the surprising discovery that feeding an imaginary number in the exponential function produced a sine wave,” (du Sautoy, 2003, p. 91). Reimann discovered that if s is an even-negative-integer that the zeta function would equal zero. He called these non-trivial zeros and didn’t concern himself too much with those. According to du Sautoy (2003, p 90), “Reimann made the stunning discovery that encoded in the varying heights of the waves was the way to correct the errors in his guess for the number of primes.” This means he was able to predict the exact number of primes less than a given number, which eliminated the error of estimating the number of primes that plagued Legendre and Gauss. He discovered this when he allowed s = 1/2+n(i). This produced a line at the point 1/2 and extended it north and south in this number plane. All values on this line produced a