It’s Crystal Clear: New Methods Emerge to Predict the Distribution Pattern of Prime Numbers

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.

Prime Time

Image from Pixabay.com

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

Fractal Rhombic Ring, Image by fdcomite

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.

Image from Pixabay.com (2)

My Questions

  • 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.

Applying Origami Folding Techniques to Strands of DNA to Produce Faster and Cheaper Computer Chips

"Origami", Image by David Wicks

“Origami”, Image by David Wicks

We all learned about the periodic table of elements in high school chemistry class. This involved becoming familiar with the names, symbols and atomic weights of all of the chemical occupants of this display. Today, the only thing I still recall from this academic experience was when the teacher told us on the first day of class that we would soon learn to laugh at the following:

Two hydrogen atoms walk into a bar and the first one says to the other “I’ve lost my electron”. The other one answers “Are you sure?”. The first one says “I’m positive.”

I still find this hilarious but whatever I recall today about learning chemistry would likely get lost at the bottom of a thimble. I know, you are probably thinking “Sew what”.

Facing the Elements

Besides everyone’s all-time favorites like oxygen and hydrogen that love to get mixed up with each other and most of the other 116 elements, another one stands alone as the foundation upon which the modern information age was born and continues to thrive today. Silicon has been used to create integrated circuits, much more commonly known as computer chips.

This has been the case since they were first fabricated in the late 1950’s. It has remained the material of choice including nearly all the chips running every imaginable one of our modern computing and communication devices. Through major advances in design, engineering and fabrication during the last five decades, chip manufacturers have been able to vastly shrink this circuitry and pack millions of components into smaller squares of this remarkable material.

A fundamental principle that has held up and guided the semiconductor industry, under relentlessly rigorous testing during silicon’s enduring run, is Moore’s Law. In its simplest terms, it states that the number of transistors that can be written onto a chip doubles nearly every two years. There have been numerous predictions for many years that the end of Moore’s Law is approaching and that another substrate, other than silicon, will be found in order to continue making chips smaller, faster and cheaper. This has not yet come to pass and may not do so for years to come.

Nonetheless, scientists and developers from a diversity of fields, industries and academia have remained in pursuit of alternative computing materials. This includes elements and compounds to improve or replace silicon’s extensible properties, and other efforts to research and fabricate entirely new computing architectures. One involves exploiting the spin states of electrons in a rapidly growing field called quantum computing (this Wikipedia link provides a detailed and accessible survey of its fundamentals and operations), and another involves using, of all things, DNA as a medium.

The field of DNA computing has actually been around in scientific labs and journals for several decades but has not gained much real traction as a viable alternative ready to produce computing chips for the modern marketplace. Recently though, a new advance was reported in a fascinating article posted on Phys.org on March 13, 2016, entitled DNA ‘origami’ Could Help Build Faster, Cheaper Computer Chips, provided by the American Chemical Society (no author is credited). I will summarize and annotate it in order to add some more context, and then pose several of my own molecular questions.

Know When to Fold ‘Em

A team of researchers reported that fabricating such chips is possible when DNA is folded and “formed into specific shapes” using a process much like origami, the Japanese art of folding paper into sculptures. They presented their findings at the 251st American Chemical Society Meeting & Exposition held in San Diego, CA during March 13 through 17, 2016. Their paper entitled 3D DNA Origami Templated Nanoscale Device Fabrication, appears listed as number 305 on Page 202 of the linked document.  Their presentation on March 14, 2016, was captured on this 16-minute YouTube video, with Adam T. Woolley, Ph.D. of Brigham Young University as the presenter for the researchers.

According to Dr. Woolley, researchers want to use DNA’s “small size, base-pairing capabilities and ability to self-assemble” in order to produce “nanoscale electronics”. By comparison, silicon chips currently in production contain features 14 nanometers wide, which turn out to be 10 times “the diameter of single-stranded DNA”. Thus, DNA could be used to build chips on a much smaller and efficient scale.

However, the problem with using DNA as a chip-building material is that it is not a good conductor of electrical current. To circumvent this, Dr. Woolley and his team is using “DNA as a scaffold” and then adding other materials to the assembly to create electronics. He is working on this with his colleagues, Robert C. Davis, Ph.D. and John N. Harb, Ph.D, at Brigham Young University. They are drawing upon their prior work on “DNA origami and DNA nanofabrication”.

Know When to Hold ‘Em

To create this new configuration of origami-ed DNA, they begin with a single long strand of it, which is comparable to a “shoelace” insofar as it is “flexible and floppy”. Then they mix this with shorter stand of DNA called “staples” which, in turn, “use base pairing” to gather and cross-link numerous other “specific segments of the long strand” to build an intended shape.

Dr. Woolley’s team is not satisfied with just replicating “two-dimensional circuits”, but rather, 3D circuitry because it can hold many more electronic components. An undergraduate who works with Dr. Woolley named Kenneth Lee, has already build such a “3-D, tube-shaped DNA origami structure”. He has been further experimenting with adding more components including “nano-sized gold particles”. He is planning to add still more nano-items to his creations with the objective of “forming a semiconductor”.

The entire team’s lead objective is to “place such tubes, and other DNA origami structures, at particular sites on the substrate”. As well, they are seeking us use gold nanoparticles to create circuits. The DNA is thus being used as “girders” to create integrated circuits.

Dr. Woolley also pointed to the advantageous cost differential between the two methods of fabrication. While traditional silicon chip fabrication facilities can cost more than $1 billion, exploiting DNA’s self-assembling capabilities “would likely entail much lower startup funding” and yield potentially “huge cost savings”.

My Questions

  • What is the optimal range and variety in design, processing power and software that can elevate DNA chips to their highest uses? Are there only very specific applications or can they be more broadly used in commercial computing, telecom, science, and other fields?
  • Can any of the advances currently being made and widely followed in the media using the CRISPR gene editing technology somehow be applied here to make more economical, extensible and/or specialized DNA chips?
  • Does DNA computing represent enough of a potential market to attract additional researchers, startups, venture capital and academic training to be considered a sustainable technology growth sector?
  • Because of the potentially lower startup and investment costs, does DNA chip development lend itself to smaller scale crowd-funded support such Kickstarter campaigns? Might this field also benefit if it was treated more as an open source movement?

February 19, 2017 Update:  On February 15, 2017, on the NOVA science show on PBS in the US, there was an absolutely fascinating documentary shown entitled The Origami Revolution. (The link is to the full 53-minute broadcast.) It covered many of the today’s revolutionary applications of origami in science, mathematics, design, architecture and biology. It was both highly informative and visually stunning. I highly recommend clicking through to learn about how some very smart people are doing incredibly imaginative and practical work in modern applications of this ancient art.