What’s your favorite number? Many people may have an irrational number in mind, such as pi (π), Euler’s number (e) or the square root of 2. But even among the natural numbers, you can find values that you encounter in a wide variety of contexts: the seven dwarfs, the seven deadly sins, 13 as an unlucky number, and 42which was popularized by the novel THE Hitchhiker’s Guide to the Galaxy by Douglas Adams.
What about a higher value such as 1729? The number certainly doesn’t sound particularly exciting to most people. At first glance, this seems downright boring. After all, it is neither a prime number, nor a power of 2, nor a square number. The numbers don’t follow any obvious pattern either. That’s what mathematician Godfrey Harold Hardy (1877-1947) thought when he got into a taxi with the identification number 1729. At the time, he was visiting his sick colleague Srinivasa Ramanujan (1887–1920) to the hospital and told him about the “boring” taxi number. He hoped it wasn’t a bad omen. Ramanujan immediately contradicted his friend“It’s a very interesting number; it is the smallest number that can be expressed as a sum of two cubes in two different ways.
Now you may wonder if there can be any number that is not interesting. This question quickly leads to a paradox: if there really is a value not which has no arousing properties, then that very fact makes it special. But there is indeed a way to determine the interesting properties of a number quite objectively – and to the surprise of mathematicians, research in 2009 suggested that the natural numbers (positive integers) fall into two well-defined camps : exciting and boring values.
A comprehensive number sequence encyclopedia provides a way to investigate these two opposing categories. Mathematician Neil Sloane came up with the idea for such a compilation in 1963, while writing his doctoral thesis. At that time, he had to calculate the height of the values in a type of graph called a tree network and came across a series of numbers: 0, 1, 8, 78, 944,… He did not yet know how to calculate exactly the digits of this sequence and would have liked to know if his colleagues had already encountered a similar sequence during their research. But unlike logarithms or formulas, there was no register for number sequences. Thus, 10 years later, Sloane published his first encyclopedia, A textbook of entire sequences, which contained about 2,400 sequences which also proved useful for performing certain calculations. The book met with enormous success: “There is the Old Testament, the New Testament and the Whole Sequence Manual“, wrote an enthusiastic reader, according to Sloane.
In the years that followed, many submissions with more sequences reached Sloane, and scientific papers with new number sequences also appeared. In 1995, this prompted the mathematician, with his colleague Simon Plouffe, to publish THE Integer Sequence Encyclopedia , which contained some 5,500 sequences. The content kept growing, but the Internet made it possible to control the flow of data: in 1996, the Online Encyclopedia of Entire Sequences (OEIS) appeared in a format free of any limitations on the number of sequences that could be recorded. As of March 2023, it contains just over 360,000 entries. Submissions can be made by anyone: a person making an entry need only explain how the sequence was generated and why it is interesting, as well as provide examples explaining the first few terms. Reviewers then check the entry and publish it if it meets these criteria.
Besides the well-known sequences such as prime numbers (2, 3, 5, 7, 11,…), powers of 2 (2, 4, 8, 16, 32,…) or the Fibonacci sequence (1 , 1, 2, 3, 5, 8, 13, …), the OEIS catalog also contains exotic examples such as the number of ways to build a stable tower from not Lego blocks studded two by four, (1, 24, 1,560, 119,580, 10,166,403,…) or the “lazy caterer sequence» (1, 2, 4, 7, 11, 16, 22, 29,…), the maximum number of pieces of pie that can be reached in not cuts.
Since approximately 130 people review submitted number sequences, and the list of such obvious candidates has been around for several decades and is fairly well known in the mathematically savvy community, the collection is meant to be an objective selection of all sequences. This makes the OEIS catalog suitable for studying the popularity of numbers. Thus, the more often a number appears in the list, the more interesting it is.
This is at least the opinion of Philippe Guglielmetti, who runs the French blog Dr Goulu. In an article, Guglielmetti recalled a former mathematics professor’s assertion that 1548 was an arbitrary number with no particular property. This number actually appears 326 times in the OEIS catalog. An example: it is displayed as a “possible period of a single cell in rule 110 cellular automaton in a cyclic universe of width not.” Hardy was also wrong when he called taxi number 1729 boring: 1,729 appears 918 times in the database (and also frequently on the TV show Futurama).
Guglielmetti therefore went in search of really boring numbers: those that hardly appear in the OEIS catalog, if at all. This is the case, for example, with the number 20,067. In March, it is the smallest number that does not appear in any of the many stored number sequences. (This is simply because the database only stores the first 180 or so characters of a sequence of numbers, otherwise every number would appear in the list of positive integers in the OEIS.) So the value 20067 seems quite boring. On the other hand, there are six entries for the number 20,068, which follows it.
But there is no boring universal law of numbers, and the status of 20,067 may change. Perhaps while writing this article, a new sequence was discovered in which 20,067 appear among the first 180 characters. Nevertheless, OEIS entries for a given number are suitable as a measure of that number’s interest.
Guglielmetti then output the number of all entries in order for natural numbers and plotted the result graphically. He found a scatter plot in the form of a wide curve that slopes towards large values. This is not surprising since only the first members of a sequence are stored in the OEIS catalog. What is surprising, however, is that the curve consists of two bands which are separated by a conspicuous gap. Thus, a natural number appears particularly frequently or extremely rarely in the OEIS database.
Fascinated by this result, Guglielmetti turned to the mathematician Jean-Paul Delahaye, who regularly writes popular science articles for For science, American scientistFrench-language sister publication. He wanted to know if experts had already studied this phenomenon. This was not the case, so Delahaye took up the subject with his colleagues Nicolas Gauvrit and Hector Zenil and delved into it. They used results from algorithmic information theory, which measures the complexity of an expression by the length of the shortest algorithm describing the expression. For example, an arbitrary five-digit number such as 47934 is harder to describe (“the sequence of digits 4, 7, 9, 3, 4”) than 16384 (214). According to an information theory theorem, numbers with many properties usually also have low complexity. That is, values that appear frequently in the OEIS catalog are most likely to be simple to describe. Delahaye, Gauvrit and Zenil were able to show that information theory predicts a similar trajectory for the complexity of natural numbers as that shown in the Guglielmetti curve. But that doesn’t explain the gaping hole in that curve, known as the “Sloane gap,” after Neil Sloane.
The three mathematicians suggested that the discrepancy stems from social factors such as a preference for certain numbers. To back this up, they ran what’s called a Monte Carlo simulation: they designed a function that maps natural numbers to natural numbers – and does it in such a way that small numbers are produced more often than large ones. The researchers put random values into the function and plotted the results against their frequency. This produced a fuzzy, sloping curve similar to that of the OEIS catalog data. And just as with information theory analysis, there is no trace of deviation.
To better understand how the gap occurs, we need to look at which numbers fall into which band. For small values up to about 300, the Sloane deviation is not very pronounced. It is only for large numbers that the gap opens significantly: about 18% of all numbers between 300 and 10,000 lie in the “interesting” band, while the remaining 82% belong to the values “boring”. It turns out that the band of interest comprises about 95.2% of all square numbers and 99.7% of prime numbers, as well as 39% of numbers with many prime factors. These three classes already represent nearly 88% of the band of interest. The remaining values have striking properties such as 1111 or the formulas 2not + 1 and 2not – 1, respectively.
According to information theory, the numbers that should be of particular interest are those that have low complexity, that is, they are easy to express. But if mathematicians consider some values more exciting than others of equal complexity, this can lead to Sloane’s deviation, as Delahaye, Gauvrit and Zenil argue. For example: 2not + 1 and 2not +2 are equally complex from an information-theoretic point of view, but only the values of the first formula are in the “interesting band”. Indeed, these numbers make it possible to study prime numbers, which is why they appear in many different contexts.
So the separation into interesting and boring numbers seems to stem from the judgments we make, such as attaching importance to prime numbers. If you want to give a really creative answer when asked what your favorite number is, you can cite a number like 20067, which doesn’t yet have an entry in Sloane’s encyclopedia.
This article originally appeared in Spektrum der Wissenschaft and has been reproduced with permission.