The Unexpected Biases of Primes
A new discovery in mathematics was actually pretty obvious.
When a new discovery is made in mathematics, generally only other mathematicians get excited by the discovery. Most new discoveries are either in unique areas of study, obscure corners of mathematics, or highly specialized fields. String theory, for example, or topology.
Because “basic” mathematics has been studied for so long by so many talented individuals, many easy-to-understand theories have been established or proven already. It doesn’t feel like there are a lot of discoveries left to be found in the areas of mathematical thought that are easily accessible to laypersons.
That’s why last year’s discovery of a Prime Number Conspiracy was stunning to so many mathematicians—it felt like an obvious result that we should have noticed before now.
A Quick Review
Prime numbers are whole numbers whose only factors are one and itself, like 7 (1 × 7) or 23 (1 × 23). Prime numbers make up a unique code for every single whole number.
Given this universal and powerful status, mathematicians have long looked for patterns within prime numbers with varying degrees of success. Certain tendencies have been found in prime numbers (by Pierre de Fermat, Christian Goldbach, and others), but usually discoveries are only along the lines of “If a prime number has this characteristic, then it also has that characteristic.”
Formulas have been found that sometimes lead to prime numbers. Mersenne’s famous formula states that 2p–1, where p is a prime, sometimes leads to another prime number (but there are no guarantees, and many prime numbers are not found using this formula). Euclid long ago proved that there are infinite prime numbers and it’s long been believed that there are infinite “twin primes” (primes that are consecutive odd numbers like 29 and 31 or 41 and 43), though a proof of that conjecture remains elusive.
Despite these historic findings, some seemingly inevitable and obvious discoveries have remained hidden. There is no formula to use to determine if a very large number is prime. There is no formula to determine the “next” prime number, or how many prime numbers might exist below a certain number (there’s a logarithmic formula that only estimates this quantity). Generally, it had been agreed that primes behave randomly and unpredictably.
Every so often, news about prime numbers pops up on the wire services. Frequently, it is because mathematicians have found the “next” largest known prime number (the current record holder has more than twenty-two million digits). In 2013, little known mathematician Yitang Zhang proved a groundbreaking result that mathematicians feel is the first step toward finalizing the “twin prime” proof mentioned earlier.
And in 2016, Robert Lemke Oliver and Kannan Soundararajan discovered something “really, really bizarre” that “floored” fellow mathematicians.
The Conspiracy
All prime numbers with two or more digits end in 1, 3, 7, or 9. Due to their unpredictability, prime numbers behave in so many ways like a group of random numbers, so we’d expect the final digit of the consecutive primes to be randomly spread out.
For example, for all primes that end in 9, we’d expect around 25% of each of the following primes to end in 1, 3, 7, and 9, respectively. That’s how randomness works. But Soundararajan and Oliver discovered that this is far from the case for the first billion primes.
Consecutive primes tend not to end in the same digit. A prime that ends in 9 is 65% more likely to be followed by a prime that ends in 1 than another prime that ends in 9. Similar percentages hold true for each of the other ending digits of prime numbers. For centuries, the assumption was that primes behave randomly, and they just don’t.
At all.
Clearly this discovery was only possible in the age of computing, which allowed mathematicians to consider the final digits of hundreds of millions of primes. That would not have been possible in the age of paper-and-pencil computations—the size of prime numbers gets unwieldy in a big hurry. Still, given the computing power, this was a relatively simple exercise in counting and sorting. A middle schooler can understand the results. So why had mathematicians not discovered this before 2016?
When considering this question, I remembered a quote from one of my favorite novels, The Curious Incident of the Dog in the Night-Time: The main character Christopher, a math savant, says that “. . . prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life.”
This isn’t a very optimistic view of life, but it’s one I’m frequently tempted to adopt. Much like prime numbers, life is complex, unpredictable, and seemingly full of uncertainty. Just as prime numbers make up the basis of our entire system of numbers, my Christian faith makes up the basis for everything else I believe.
But my Christian faith at times feels unstable, shaky, random, and chaotic. I feel at times like I am expected to have faith in situations and circumstances where I just don’t see any practical reasons to have faith. What is supposed to be foundational and supportive in my life can at times feel as random as mathematicians long believed prime numbers to be.
Looking for Simplicity
How do I typically respond in situations of doubt and uncertainty in my faith? I go cosmic. I look for a big-picture, comprehensive understanding of my situation. Like the mathematicians who studied prime numbers, sometimes this thought process leads to success and I discover something useful about myself, my faith, or my Creator.
A Mersenne formula for my life, if you will—a formula for my beliefs that works on occasions, but just as frequently doesn’t lead to success. Just as often as the formula works, I end up completely frustrated; the big-picture view for my life seems exasperatingly out of reach, and I’m stuck with a random collection of events that feels like it should have a pattern or predictability to it but just doesn’t.
Mathematicians overlooked a seemingly obvious trait of prime numbers for decades. While they were looking for an overarching formula that predicted prime numbers, proof to some existing conjectures, or had given up on the topic entirely, prime numbers were quietly whispering the entire time, “Psst. We don’t like to repeat our final digit. We don’t behave randomly.”
Who knows what discoveries this could lead to in the future? It may lead to very little. It may lead to some exciting new bridge between number theory and the world of quantum physics. There’s a lot of work to be done, but the discovery has given a starting point to study prime numbers from an entirely new perspective. Sometimes the most interesting and unexpected discoveries occur on a much simpler level than where we were originally looking for answers.
The same can be true in my faith. While I’m casting about the Bible looking for a verse that lays out the meaning of my entire life, God whispers, “Psst. You are loved. Rest in that a while.” That relatively simple answer may be all I need to get a new perspective on the doubts and confusion in my life. It’s a new path to solutions, giving a new approach to the questions I face.
Christopher was right: Prime numbers are like life. We’re still learning about them both, and if we’re willing to look for simpler solutions, we might be surprised at what we can learn.
Cover image by Roman Mager.
