We encounter our first mathematical pattern when we learn how to count: 1, 2, 3, 4, 5 and so on. This is just one example of a sequence, a list of numbers arranged in a particular order. It’s also one of the most fundamental, bare-bones objects in math. If you think you have zero mathematical intuition, just look at the sequence 1, 3, 5, 7, and you can likely guess what comes next. The accessibility of these kinds of questions has drawn in recreational mathematicians for decades or longer.
But despite their simplicity, sequences also encode deep mathematical relationships. By probing sequences, mathematicians have made surprising discoveries that have greatly influenced the history of the field.
The twin motivations of pure research and pure fun collide beautifully in the On-Line Encyclopedia of Integer Sequences (OEIS), a database of sequences created by Neil Sloane in 1964. (Erica Klarreich profiled Sloane for Quanta in 2015.) The OEIS is a trove of whimsical oddities that mathematicians have come across while toying with numbers in their free time. But it also contains some of the oldest and most important objects in math.
Today, lists of numbers continue to intrigue and flummox number theorists, while offering new insights into problems all over mathematics. Let’s take a look at a few of them and see what makes them so interesting.
Sequence of Events
Consider the sequence formed when you take every whole number, square it, and add 1. You get 2, 5, 10, 17 and so on. Though straightforwardly defined, the n2 + 1 sequence (A002522 in the OEIS) is actually very difficult to study, because it combines multiplication (squaring a number) with addition (adding 1). Both operations are simple enough on their own, but when combined, they lead to deep mathematical questions. In fact, the strange relationship between addition and multiplication forms one of the biggest tensions in number theory — and the n2 + 1 sequence provides a perfect entry point for studying it. Last year, Quanta reported on the latest advance in understanding the sequence: The mathematician Hector Pasten proved that all of its numbers have a relatively large prime factor. The work also marked the first progress in a while on a particularly famous problem about addition and multiplication, one rife with drama — the abc conjecture.
Other kinds of sequences are inspired by nature. Take the Fibonacci sequence (A000045), which is what mathematicians call recursive: You can always calculate the next number using those that come before it. In this case, each number is the sum of the two preceding numbers in the list, so that you get 0, 1, 1, 2, 3, 5, 8, etc. This sequence, which was first described in Indian poetry dating back millennia, appears all over nature — in the arrangement of a pine cone, in the syllables of human speech, in the population dynamics of certain species.
This sequence is also known to pop up unexpectedly all over math. In 2022, for instance, mathematicians found the Fibonacci numbers hidden in a host of geometric structures. The following year, another team proved that the solutions to a famous equation are related to each other in part through the Fibonacci sequence (as well as through another ancient, recursive list of numbers, called the Pell sequence [A000129]). New discoveries about the Fibonacci numbers continue to pile up on a weekly basis.
Another kind of sequence that shows up all over math: lists of evenly spaced numbers called arithmetic progressions. For example, the sequence 2, 5, 8, 11, 14, etc. (A016789), is an arithmetic progression whose numbers all differ by 3. Quanta has covered several results in which mathematicians study how arithmetic progressions inevitably appear in random sets of numbers. In this way, sequences are a great way to explore how, no matter how hard you try, you can’t avoid the emergence of mathematical order.
But the most important sequence for research mathematicians is the one formed by the prime numbers (A000040), the building blocks of all other numbers. It’s still a mystery what numbers even belong to this sequence: As the number line stretches to infinity, it becomes harder and harder to pinpoint where the primes are located. Check out this earlier edition of Fundamentals to learn more about prime numbers.
These are just a few of the many sequences that mathematicians enjoy studying. Researchers will likely never run out of simple but hard-to-answer questions concerning ordered lists of numbers. In this way, sequences will continue to be a source of crucial discoveries — and entertainment, as evidenced by the OEIS’s enduring popularity more than half a century after its creation. |
