Maths Club: Paul Erdős

Paul Erdős and the INFINITUDE of  the PRIMES

– from the annals of MathsClub of Science Circle

Presentation by Tagline – Robert A. Hendrix, M.D.

Great progress in mathematics was achieved during the tumultuous 20.century. There was such a plethora of new, conceptual mathematics that of necessity,

• there is always a great deal they never get around to teaching you in your PhD programs regardless of your discipline.

Among other things, it was the century of the eccentric, Budapest-born Paul Erdős (pronounced “Pal Air-dush”, b. 26 March 1913 – d. 20 September 1996). Over his lifetime, Erdős became the most published mathematician in history (around 1,500 mathematical papers as well as six books). We will take a close look at the life lived by this man – this unusual, “homeless” mathematician, described as the “MAN WHO LOVED ONLY NUMBERS”. 

And we will consider the breadth of his contributions to modern mathematics.

A series is a countably INFINITE SUM of numerical terms. We will define how such a series can have a definite total, in which case it CONVERGES, or if not, then it DIVERGES. Firstly, we will reflect on convergence vs divergence of a few infinite series; secondly, a quick look at the Harmonic Series;  then we will focus on the question of CONVERGENCE of the “HARMONIC SERIES OF THE PRIMES“:

Now ask yourself, “Does this infinite series actually converge? Does it actually equal a real number?”

This question was first settled by Swiss mathematician, Leonhard Euler around 1740, but his proof involved analytic functions such as the natural logarithmic function, as well as some “sleight of hand” using intuitive mathematics that “crossed the line”.  Paul Erdős has an extraordinary and clever proof that further settles this question, and his proof does NOT call on any special or analytic functions, but rather depends on pure, indisputable logic based on a deep dive into the properties of numbers.  Consequently, the very approachable Erdős proof is a CLASSIC – we will deconstruct this proof step-by-step such that any audience member with a cup of coffee in hand can follow it. Erdős himself stated “a mathematician is a machine that transforms coffee into equations.”

Once you see the brilliance and depth of Erdős proof, you will come away with a sense of awe for the depth and cleverness of pure number theory. 

We will also discuss the INFINITUDE OF THE PRIMES and EUCLID NUMBERS as a subset of Natural (counting) Numbers. Finally, we will discuss the Ancient Greek LIMITATIONS in their understanding of NUMBERS. In closing, we will state a couple of UNSOLVED PROBLEMS IN MODERN MATHEMATICS for members of the Science Circle audience to work on.