Unexpected Beauty in Primes

The Visualization of Primes & Its Potential Significance

The significance of prime numbers, in both everyday applications & as a subtopic pertinent to all branches of math, cannot be overstated . We quietly rely on their special properties to carry the backbone of countless parts of our society — all because they are an irreducible part of the very fabric of nature. Resistant to any further factorization, prime numbers are often referred to as the “atoms” of the math world. As Carl Sagan so eloquently describes them:

There’s a certain importance to prime numbers’ status as the most fundamental building blocks of all numbers, which are themselves the building blocks of our understanding of the universe.

The use of prime numbers in nature & in our lives are everywhere: cicadas time their life cycles by them, clock-makers use them to calculate ticks, & aeronautical engines use them to balance frequency of air pulses. However, all of these use-cases pale in comparison to the one fact every cryptographer is familiar with: prime numbers are at the very heart of modern computational security, which means prime numbers are directly responsible for securing pretty much everything. See that lock in the URL bar? Yeap, a two-key handshake powered by primes. How is your credit card protected on purchases? Again, encryption powered by primes.

Yet for our consistent reliance on their unique properties, prime numbers have remained infamously elusive. Throughout the history of math, the greatest minds have attempted to prove a theorem for predicting which numbers are prime, or, how apart successive primes are in placement. In fact, a handful of unsolved problems such as Twin Primes , Goldbach Conjecture , Palindromic Primes , & The Riemann Hypothesis all revolve around this general unpredictability & uncertainty in prime numbers as they approach infinity. Granted, since the early days of Euclid we’ve found a handful algorithms that predict some placement, but general theorems haven’t been accepted nor did previous attempts have the tools to test large numbers. 21st-century technology, however, does allow researchers to test proposals with extremely large numbers, but that method alone invites controversy as brute-force testing isn’t quite globally accepted as a solid proof. In other words, primes have resisted any universal formula or equation, their appearance in nature remaining a status of seemingly-random.

As It Turns Out However, A Simple Scribble Proves That They’re At Least Not Completely Random…

Scribbling To A Clue — The Ulam Spiral

One of the greatest proofs we have that the appearance of prime numbers is no mere coincidence came from one of the most unlikeliest places: the effortless & accidental doodles of one bored lecture attendee.

Ulam Spiral Setup

As the story goes, one Polish mathematician, Stanislaw Ulam , stumbled upon a visual pattern in 1963 during a seminar. While drawing a grid of lines, he decided to number the intersections according to a square-spiral pattern, & then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam’s slightly more formal prose, it:

Appears To Exhibit A Strongly Nonrandom Appearance

The Ulam Spiral, or prime spiral, is the resultant graphical depiction from marking the set of prime numbers in a square-spiral. It was originally published & reached the mainstream through Martin Gardner’s Mathematical Games column in Scientific American .

377x377 (~142K) Ulam Spiral

The visualization above clearly highlights significant patterns, especially diagonally. But maybe we’re deceiving ourselves? A common counter to the Ulam spiral is that perhaps our brain is simply tricking us into assigning these patterns in randomness. Luckily, we can take two different approaches to confirm that this isn’t the case. Both a visual comparison & a logical walk-through will convince one that they’re certainly not random. First, we’ll compare an Ulam spiral made by a matrix of NxN dimensions, to an equally-sized NxN matrix containing randomly-assigned dots. Second, we’ll flex our knowledge of polynomials to reason through exactly why we should expect some pattern in the visual layout of primes.

As mentioned, the likely most intuitive way to confirm a non-random pattern with a Ulam spiral is through a direct comparison. This is done by creating both a Ulam spiral & a spiral with random placements of the same size. Below are two different 200x200 matrices that represent numerical spirals: