Despite the unsettling effects of figuring out fractions, performing long division and taking the square roots of numbers have had for an eternity upon many students in the middle grades, some of these people continue on to adulthood with at least an appreciation of what mathematics can do in the real world. While they might still break into a slight sweat if challenged to quickly calculate the equivalent of 3/8, they still realize the importance of doing so and, moreover, applying that value to solve a problem.
So too, just as math teachers everywhere exhort their students to “put on their thinking caps”, sometimes a math story appears in the news that takes a bit more concentration to fully comprehend, but nonetheless really does have a certain technological cool and practicality to it. What is equally intriguing is when such a new development has the potential to eventually impact other areas of innovation that appear at first to be disparate or even unrecognizable. On its face, scientific advance X could not possibly be related to mathematical outcome Y until, by virtue of some very unconventional thinkers in another field, the real possibility emerges of a workable application of X to achieve Y.
Let’s take our virtual calculators out of their pocket protectors and have a look at such a recent advancement that is not only useful as party fun for math geeks. Rather, it may have meaningful significant in encryption science and, in turn, online security, e-commerce and data privacy. This achievement was reported in a fascinating article entitled Researchers Discover a Pattern to the Seemingly Random Distribution of Prime Numbers, by Liv Boeree, posted on Motherboard.com on September 14, 2018.
I will summarize and annotate this, and then pose several of my own equation-free questions.
First, the basics: Prime numbers (“primes”) are whole numbers that are only divisible by 1 and themselves. They start out small as 2, 3 and 5 and range upwards towards infinity.¹ As these primes are plotted out along on a graph they appear to be increasingly random with no discernible or predictable pattern.
Nonetheless, one of the greatest unsolved math problems is called the Reimann Hypothesis which, among its other brain-bending complexities, posits that there may well be a pattern to the distribution of primes but it has not yet been derived.² Discovering such a pattern would be a monumental accomplishment with major significance in mathematics, physics and modern cryptography, the latter of which is based upon large prime numbers. (More about this below).
Recently, three researchers at Princeton University have made such a discovery about an atomic pattern in a physical material comparable to the distribution of primes. They have found similarities involving primes and “certain naturally occurring crystalline materials”.³ Their recent scientific paper detailing this work is entitled Uncovering Multiscale Order in the Prime Numbers Via Scattering, by Salvatore Torquato, Ge Zhang and Matthew de Courcy-Ireland, was published in the Journal of Statistical Mechanics: Theory and Experiment, on September 5, 2018.
The unpredictability of finding new primes is not always necessarily a detriment. For example, modern cryptography methods such as the RSA encryption algorithm depends upon this factor when it comes to very large primes. This relies upon the principle that it is simple enough to take two large prime numbers and multiply them but intensely difficult to reverse this in an effort to determine exactly which two primes were used.
[While this post was being drafted, an article was posted on BusinessInsider.com on September 25, 2018 entitled An Eminent Mathematician Claims to Have Solved One of Math’s Greatest Mysteries — and It’s One of 6 Problems With a $1 Million Prize, by Andy Kiersz, reported that Sir Michael Atiyah has achieved a solution to the Reimann Hypothesis. However, this remains to be vetted by other mathematicians in this field. This problem is one of six remaining great unsolved math problems, termed the “Millennial Problems”, for which the Clay Mathematics Institute has offered a $1 million prize for the solution to each.4 This article also contains concise descriptions of the other five problems.]
Fine Crystal Settings
In a process known as X-ray diffraction, chemists and physicists study the atomic structure of a material by exposing it to x-rays and observing how the beams “scatter off the atoms within it”. Different materials will produce a variety of such patterns and indicate “how symmetrically their atoms are arranged”. In the case of a crystal, whose atomic structure is more firm than other materials such as liquids, the x-ray’s pattern of diffraction is “more orderly”.
In 2017, the lead author of the paper, Professor Salvatore Torquato, wondered whether primes could be “modeled as atom-like particles” and whether they would also form a pattern. Along with his co-authors, together they “computationally represented the primes as a one-dimensional string of atoms” and then “scattered light off them”.
They found that this created a “quasicrystal-like inference pattern” that was also a previously unseen form of fractal pattern termed “hyperuniformity“. It is exhibited by only a several “materials and systems in nature”. Included among them are prime numbers. This finding might turn out to be useful in studying such non-repeating patterns in a new field of research called “aperiodic order“.
Professor Torquato said in an article in Quanta Magazine entitled A Chemist Shines Light on a Surprising Prime Number Pattern, by Natalie Wolchover, dated May 14, 2018, that there is a resulting implication that primes “are a completely new category of structures” when viewing them as a form of physical system.
Much of the interest surrounding the new paper is its “unique intersection between the physical and more abstract mathematical realms”. As well, it contains a new algorithm that permits the prediction of primes “with high accuracy”. In time this may prove to be another advance in decisively solving the mysteries of the primes.
- If Professor Torquato’s and his co-authors’ paper and algorithm prove to be genuinely able to predict the patterns of the appearance of primes, does this actually strengthen and/or weaken the foundation of RSA-based encryption?
- Moreover, if Sir Atiyah’s has, in fact, solved the Reimann Hypothesis, what are the potential positive and negative effects upon the whole field of cryptography? Are there any additional impacts on other fields of science, math, physics and technology?
- If and when practical quantum computing becomes a reality and results in the capability to much more rapidly factor primes used in encryption, how will the work of Professor Torquato and Sir Atiyah be affected?
- So, how much is 3/8 anyway?
1. Currently, the largest prime number ever discovered was identified in 2017 and has 23,239,425 digits. That’s a lot.
2. For an outstanding history of the pursuit of prime numbers and the mathematical quest to discover a pattern in their distribution, I very high recommend reading The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, by Marcus du Sautoy, Harper Perennial; Reprint edition (August 14, 2012). This is a very accessible and literate book that presents a variety of engaging stories and deep insights into what might otherwise have otherwise appeared to have been a rather dry subject.
3. A more technical report on this story was posted on Princeton’s website entitled Surprising Hidden Order Unites Prime Numbers and Crystal-like Materials, by Kevin McElwee, on September 5, 2018.
4. I suggest adding a seventh intractable problem to this list that will likely never be solved: Finding a parking spot in my neighborhood.