An exciting new mathematical discovery might have scientists around the globe working closely with crystals in a whole new way. Recently, a team of researchers from Princeton University solved a centuries-long mystery surrounding prime numbers, AKA “the building blocks of mathematics.” For hundreds of years, no one could find a pattern to prime numbers; experts thought they were entirely chaotic.
Thanks to a groundbreaking new modeling technique, though, we now know that primes have a surprising connection to “naturally occurring crystalline materials.” This groundbreaking work could make some serious waves in the worlds of physics and materials science.
PRIME NUMBERS – A REFRESHER
Prime numbers are whole numbers that can only be divided by the number one or themselves. For example, the numbers 2, 3, 5, 7, and 11 are all prime numbers. Now, these prime numbers appear along the number line in what previously seemed like an entirely chaotic way. On top of that, the higher you go on the number line, the more erratic the numbers seem to get. A British mathematician explained the randomness of prime numbers.
“It is evident that the primes are randomly distributed out, unfortunately, we do not know what ‘random’ means,” said R.C. Vaughan.
That randomness, however, was not without its uses. Modern cryptography uses the extreme unpredictability of prime numbers to its benefit. For example, an encryption algorithm called RSA uses the chaos to its benefit. The algorithm relies on the fact that it’s simple to take two large prime numbers and multiply them. However, figuring out which of the prime numbers went into the calculation is exceedingly more difficult. If you are interested in more details, click here.
Even though there were benefits to the chaos surrounding prime numbers, it was not without its problems. Which led Luckily, theoretical chemist and Princeton Professor Salvatore Torquato had a fortuitous hunch. You see, Torquato studies materials at Princeton. One of the ways that chemists and physicists study materials are by using an X-ray.
This process is called X-ray diffraction. It works by firing X-rays at the sample of the material you want to study. The image that shows up on the screen is a view of how the rays scatter off of the atoms within the material. A liquid, for example, will show that atoms are all jumbled and all over the place. In a crystal, though, there is a rigid lattice system.
If one fires an X-ray through a diamond, for example, the resulting image shows that there is a “repetitive internal structure.” Which led Torquato to the question of the day — If he modeled the prime numbers the same as atom-like particles, would they also create a pattern?
Torquato along with his student, Ge Zhang, and number theorist Matthew de-Courcy-Ireland set up a study. The team “computationally represented the primes as a one-dimensional string of atoms and scattered light off them.”
And, Eureka, the result created a quasicrystal-like pattern of interference. Not only that, but this fractal pattern is like nothing number theorists have seen before. The Journal of Statistical Mechanics: Theory and Experiment published the initial results of the study. Torquato told Quanta Magazine that when you lay out the primes as a physical system, they are, “a completely new category of structures.”
“What’s beautiful about this is it gives us a crystallographer’s view of that the primes look like,” said Henry Cohn, a mathematician at Microsoft Research New England and the Massachusettes Institute of Technology.
And suddenly scientists have to look at crystals and what they represent in a whole new way.
WHERE WE GO FROM HERE
Now, while this discovery is certainly groundbreaking in some areas, that’s not really the case for number theorists. Mathematicians have already done all the related calculations in other forms, you see. There is a new field of study, though, called “aperiodic order.” Aperiodic order is the study of non-repeating patterns. This research could prove hugely beneficial to the people working in that field.
Henry Cohn said it best when he told Princeton,
“It’s a beautiful new perspective on this information, and it opens up new connections with materials science and scattering theory.”
And if that wasn’t enough, there’s even good news for students. The paper claims that there is a new algorithm as a result of this study that, “enables one to predict primes with high accuracy.” One thing is certain, even though the usefulness of this information is still relatively unknown, this is a huge leap toward understanding prime numbers and “solving the enigma,” that surrounds them.