I have always been fascinated with “origins”: the origin of life on Earth, the origin of human beings, the origin of the Universe, and of course to fundamental particles. I reserve a sort of special place to a mathematical entity that some people regard as fundamental: prime numbers.
A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Numbers that have more than two factors are called composite numbers.WhatIs.com
The oddity (pun intended) of these numbers attracted the curiosity of mathematicians for centuries. For example, today we still employ a simple method to tell whether a given number is prime is called the Sieve of Eratosthenes, who was a Greek mathematician known for having measured the Earth radius in 240 BC. To find all the prime numbers less than or equal to a given integer n:
- Create a list of consecutive integers from 2 through n: (2, 3, 4, …, n).
- Let p = 2, the smallest prime number.
- Enumerate the multiples of p by counting in increments of p from 2p to n, and mark them in the list (these will be 2p, 3p, 4p, …; the p itself should not be marked).
- Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now equal this new number (which is the next prime), and repeat from step 3.
When the algorithm terminates, the numbers remaining not marked in the list are all the primes below n. In the following centuries, the interest in this class of numbers did not decline. In 1640 Pierre de Fermat stated the Little Theorem (not to be confused with Fermat’s Last Theorem):
If p is a prime and a is any integer not divisible by p, then a^(p − 1) − 1 is divisible by p.
Leonhard Euler provided the first published proof in 1736, but Gottfired Leibniz had given basically the same proof in an unpublished manuscript fifty years before. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Euler proposing the following conjecture:
Every even integer greater than 2 can be expressed as the sum of two prime numbers
To this date, Goldbach’s conjecture is still unproven, even though it has been demonstrated to hold true up to 3,325,581,707,333,960,528 by Tomás Oliveira e Silva.
Among the many strange properties of prime numbers is the Twin prime conjecture, i.e. the observation that many primes are in fact either 2 less or 2 more than another prime number such as 11 and 13.
How many primes are there?
A related question is how many primes are there below a given number. The prime counting function π(N) is defined as the number of primes not greater than N. For example, π(10) = 4, since there are four primes less than or equal to ten (2, 3, 5 and 7). Intuitively, primes become less common as they become larger, but it is not obvious at what rate this occurs. This idea is formalized in the Prime Number Theorem. To start with, it was noted that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). So the theorem states that π(N) is equivalent to N/log(N) as N approaches infinity. However, over the years it was discovered that another function called Logarithmic Integral Function Li(N) was a better approximation to π(N).
By extrapolating the difference between π(N) and the other two functions, one can see that both converge towards π(N) at infinity, but Li(N) does that much faster.
The Riemann Hypothesis
The German mathematician Bernhard Riemann worked mainly in the field of the geometry of non-Euclidean (or curved) spaces, in which he laid the basis for what will become the mathematics of Einstein’s General Relativity. Nonetheless, he’s also famous for a conjecture that to this date is still unproven (or is it?). As a matter of fact, his only paper on number theory is also probably the most important of all times in the field! In this work, Riemann discovered the exact formula for counting the number of primes below N, i.e. π(N) ), which can be expressed as follows:
The function ζ(s) is a complex function usually referred to as Riemann Zeta function. The Riemann hypothesis states that the zeros of ζ(s) (i.e. the numbers s for which ζ(s) = 0) are all either negative even numbers, or complex numbers with real part equal to 1/2. Riemann discovered that the exact expression of π(N) involves the roots of the zeta function. The Riemann conjecture is part of the 23 Hilbert’s problems and its proof is still up for grabs!
In the 1970s, some physicists such as Freeman Dyson and Hugh Lowell Montgomery speculated that the zeros of ζ(s) may play some role in quantum mechanics. The Primon gas is an example of a quantum field theory of a set of non-interacting particles whose states depend on prime numbers. It can be shown that the Hamiltonian function describing the energy of such a system is related to the Riemann Zeta function. In some sense, there may be a chance that Nature has found a way by itself to prove the Riemann Hypothesis!
The Prime Number Cross
Many years ago I have spent a few days in a hospital. I asked my parents to buy me a book called God’s Secret Formula: The Deciphering of the Riddle of the Universe and the Prime Number Code by the German Chemist Peter Plichta. I’m not sure I would recommend reading it, but it certainly contains something that kept bugging me over the years. That thing is a way to organize natural numbers in concentric rings of 24 numbers each. I’m not sure where he got the inspiration from, my best guess is that he started somehow from the distribution of twin primes. Here’s what it looks like for numbers up to 96:
The overall idea is that, apart from two exception on the first ring (integers 2 and 3), all the other prime numbers lay on 8 radii, but not all numbers that appear on those lines are primes. Also, the radius stemming from 1 contains the squares of prime numbers larger than 5, e.g. 5*5=25, 7*7=49, 11*11=121, 13*13=169, 17*17=289 and so on…
The authors speculates that this distribution resembles the Maltese Cross, an observation that might indicate that the Knights Hospitaller might have known that since a very long time. Furthermore, he insists that this geometrical arrangement may have some deeper implication in nuclear physics, a claim that I believe is utter nonsense.
In any event, I wrote a python code based on numpy and matplotlib to create this kind of figures. You can grab the code from my GitHub repository.
A Deeper Pattern?
To conclude, the study of prime numbers is very fascinating, but it’s unclear to me whether cracking the code of the primes will really give us a clearer understanding of mathematics, or if it will remain confined to the realm of mere curiosities. Not being a mathematician myself, I find hard to grasp whether the Prime Number Cross can be used to further investigate (or even prove!) some of the unsolved conjectures discussed above. In any event, feel free to play around with the python code and try your chance to win the next Fields Medal.