SOME INTERESTING MATHEMATICAL TOPICS

Tom Kando

Most of my recent posts have been negative. There are so many wrong things these days. The third year of Covid, 5th month of war in Ukraine, the Supreme Court abolishing our constitutional rights, the proliferation of weapons and therefore of mass murder, no progress on the environmental front, the rightward political drift toward fascism in many countries, etc. 

So how about something not depressing, for once? 

A field which has long piqued my curiosity is mathematics, if only because I am not well versed in that area. Incidentally, some of my ancestors were eminent mathematicians and scientists. They include my great-grandfather Beke Mano, who was a pioneer in differential equations and my grand-uncle 

Kalman Kando who invented the phase converter. 

My secondary school education was stellar on the humanities side, but mediocre on the quantitative side: At the gymnasium, we had six years of six languages - Dutch, English, French, German, Latin and Greek! However, our quantitative training did not go beyond algebra, trigonometry, analytic geometry and stereometry. Later, obtaining my PhD at the University of Minnesota required a strong quantitative component in the form of advanced statistics. However, most of my quantitative skills, limited to begin with, have atrophied. I remain fascinated by fields about which I know little, wondering sometimes if I might chose the direction of the exact sciences if I were to do things all over again. 

Take prime numbers, for instance. You can check out a previous post of mine about this: 

Prime Numbers, Large Numbers 

A prime number is a whole number that can only be divided by 1 and by itself. Or put differently, prime numbers cannot be divided by any other whole number without leaving a fraction. The smallest twenty-five prime numbers (those under 100) are: 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97. 

I won’t go into how you go about finding out whether a number is a prime, but to say that you have to factorize it. You can Google it. 

One fascinating characteristic of prime numbers is that they seem to obey no other law than that of chance, and that nobody can predict what the next higher - and not yet discovered - prime number will be. 

Over the centuries, mathematicians have discovered larger and larger prime numbers, and now with computers, the search has reached astronomical levels. 

The unfathomable magnitude of the prime numbers which current researchers are looking for - and finding - totally blows my mind: As of 2022, the largest prime number ever discovered (by the Greater Internet Mersenne Prime Search Research Project) was 2 to the power 82,589,933 minus 1. This number has nearly 25 million digits. To write it out would require 8,500 pages, or twenty-eight 300-page long books. see: Largest know prime number and Prime Number. Researchers are currently looking for primes with 1 billion digits. If they discover one, writing out that number would require 350 thousand pages, or 1,200 books. 

As you get into large numbers, factorization becomes tediously time-consuming. But there are many prime number calculators on the Internet, for example: Prime Numbers Calculator, which you can use up to nearly four and a half billion. 

Another fascinating aspects of prime numbers: There is an INFINITE amount of them, but at the same time, they become more and more rare as you move up the sequence of natural numbers. This is something you suspect right away, even just looking at the first few hundred numbers: For example, here is how many prime numbers there are under 500: 

from 1 to 100: 25 

101 to 200: 21 

201 to 300: 16 

301 to 400: 17 

401 to 500: 16 

The diminishing number of primes is called the Prime Number Theorem (PNT). Mathematicians have known this for hundreds of years. For example, Carl Friedrich Gauss - of the famous Gaussian (normal) distribution - studied and confirmed this theorem in 1791, at the age of 14. 

Mathematicians say that “the distribution of primes is ASYMPTOTIC.” This means that the curve approaches zero as it tends towards infinity. In other words, it never touches zero, and its slope is curvilinear, declining ever more slowly. See: Asymptotic Curve 

I just wanted to SEE the asymptotic distribution of prime numbers with my own eyes, so I examined some tables showing the density of primes as you go up the sequence of natural numbers. Here is what I found: 

below 10, there are 4 prime numbers = 40 per 100 

below 100: 25 = 25 per 100 

below 1,000: 138 = 14 per 100 

below 10,000: 1,229 = 12 per 100 

below 100,000: 9,592 = 10 per 100 

below 1 million: 78,498 = 8 per 100 

below 10 million : 664,579 = 7 per 100 

below 100 million: 5,761,455 = 6 per 100 

below 1 billion: 50,847,534 = 5 per 100 

below 10 billion: 455,052,511 = 4.6 per 100 

 

So it’s clear that prime numbers become more rare as you go up the series of natural numbers. Also, the DECLINE in the frequency of prime numbers slows down. 

Mathematicians say that the frequency reaches zero in infinity or at the “limit.” In other words, there NEVER comes a number, no matter how unimaginably large, beyond which there are zero primes, or only one, or only a finite amount of them, even though they are spaced further and further apart, which is what makes finding them increasingly difficult, even with the most powerful computers. 

I find it incredibly weird that there are immense numbers out there which are indivisible, in other words, quantities of just ONE piece, without ANY component parts. 

My next post will deal with another fascinating mathematical problem: Pi ( π).

  leave comment here

© Tom Kando 2022;All Rights Reserved