what’s your favorite number? Pi (π) and Euler number (e) or the square root of 2. But even among the natural numbers, we can find values that we encounter in different contexts, such as the seven dwarfs, the seven deadly sins, 13 as an ominous number, and 42 as a generalization.according to the novel of The Hitchhiker’s Guide to the Galaxy By Douglas Adams.
What about large values like 1,729? This number certainly doesn’t seem particularly exciting to most people. At first glance, it looks really boring. After all, it’s not a prime number, a power of 2, or a square. Also, the numbers do not follow any obvious pattern. Mathematician Godfrey Harold Hardy (1877–1947) thought this when he got into a cab with identification number 1729. Boring ‘taxi number. He hoped it was not a bad omen. This is the smallest number that can be expressed as the sum of two cubes in two different ways. “
Now you may be wondering if there could be numbers that are totally uninteresting.That question quickly leads to a paradox: is it really worth it? n It doesn’t have any exciting properties, but it’s this very fact that makes it special. And to the great surprise of mathematicians, a 2009 study suggested that natural numbers (positive integers) fall into two well-defined camps: exciting values and boring values.
A comprehensive encyclopedia of number sequences provides the means to explore these two opposing categories. Mathematician Neil Sloan came up with the idea of such compilation in 1963 when he was writing his doctoral dissertation. At the time he had to calculate the height of values in a type of graph called tree his network and came across a series of numbers 0, 1, 8, 78, 944… he still doesn’t know how to calculate didn’t know I was curious to know if his colleagues had already encountered similar sequences during their research. And ten years later Sloan published his first encyclopedia. Handbook of Integer Sequences, It contains about 2,400 sequences and has proven useful for certain computations as well. This book has received tremendous support. Handbook of Integer Sequences‘ wrote one avid reader., According to Sloan.
In the years that followed, numerous submissions containing more sequences came to Sloan, as well as scientific papers containing new number sequences. I made it a trigger. of Encyclopedia of Integer Sequences , which contains about 5,500 sequences. Content continued to grow, but the Internet made it possible to control the flood of data: In 1996, the Online Encyclopedia of Integer Sequences (OEIS) announced that there was no limit to the number of sequences that could be recorded. appeared in the form As of March 2023, it contains over 360,000 entries. Contributions can be made by anyone: the person making the entry explains how the sequence was generated and why it is interesting, as well as providing an example explaining the first few terms. Reviewers will then check the entry and publish it if it meets these criteria.
Prime numbers (2, 3, 5, 7, 11, …), powers of 2 (2, 4, 8, 16, 32, …), Fibonacci sequences (1, 1, 2, 3, 5, 8 , 13,…), the OEIS catalog also contains quirky examples such as a number of ways to build a stable tower. n 2 x 4 studded LEGO bricks (1, 24, 1,560, 119,580, 10,166, 403,…) or “lazy catering sequence” (1, 2, 4, 7, 11, 16, 22, 29, . ..) ), the maximum number of pie pieces that can be achieved by n cut
About 130 people have reviewed the submitted sequences, and lists containing these obvious candidates have existed for decades and are so well known in the mathematics-savvy community that the collection is an objective view of all sequences. This makes the OEIS catalog a good place to explore the popularity of numbers. Therefore, the more often a number appears in the list, the more interesting it is.
At least that was the idea of Philippe Guglielmetti, who runs the French-language blog Dr. Goulu. In one post, Guglielmetti recalled a former math teacher’s assertion that 1,548 is an arbitrary number with no special properties. This number actually appears 326 times in his OEIS catalog.One example: it appears as “the final period of a single cell of a rule-110 cellular automaton in a periodic universe of widths” nHardy was also wrong in calling taxi number 1729 boring. futurama).
So Guglielmetti went looking for really boring numbers. It is a number that is rarely listed in the OEIS catalog. The latter, for example he is the number 20,067. As of March, it is the smallest number that does not appear in any of the many stored number sequences. (This is because databases only store the first 180 or so characters of a number sequence; otherwise, all numbers would appear in his OEIS list of positive integers.) So the value 20,067 is very Sounds boring. In contrast, the number 20,068 that follows him has six entries.
But there are no universal laws of boring numbers, and the status of 20,067 can change. Perhaps while writing this article, a new sequence was discovered in which 20,067 of his first 180 characters appear. Nonetheless, the OEIS entry for a particular number is a good measure of how interesting that number is.
Guglielmetti printed the number of all the entries of the natural numbers in order and plotted the results on a graph. He finds a cloud of points in the shape of a broad curve that slopes towards large values. As long as only the first member of the sequence is stored in his OEIS catalog, this is not surprising. What is surprising, however, is that this curve consists of two bands separated by a clearly visible gap. Natural numbers therefore appear particularly frequently or very rarely in his OEIS database.
Fascinated by this result, Guglielmetti consulted mathematician Jean-Paul Delhaet. about science Scientific AmericanFrench version sister publication. He wanted to know if experts were already studying the phenomenon. This was not the case, so Delahaye took up the topic with his colleagues Nicolas Gauvrit and Hector Zenil to investigate it more closely. They used results from algorithmic information theory, which measures the complexity of an expression by the length of the shortest algorithm that describes the expression. For example, any 5-digit number such as 47,934 becomes 16,384 (214). According to a theorem of information theory, numbers with more properties usually also have less complexity. This means that values that appear frequently in the OEIS catalog are likely to be the easiest to describe. Delahaye, Gauvrit, and Zenil were able to show that information theory predicts a trajectory for the complexity of the natural numbers similar to that shown in Guglielmetti’s curve. But this doesn’t explain the gaping hole in that curve, known as “Sloan’s Gap” after Neil Sloan.
Three mathematicians suggested that the gap stems from social factors such as preference for certain numbers. To demonstrate this, they ran what they called a Monte Carlo simulation. They designed a function that maps natural numbers to natural numbers, so that small numbers are output more often than large numbers. The researchers put random values into the function and plotted the results according to frequency. This produced a fuzzy, sloping curve similar to the data in the OEIS catalog. And, as with information theory analysis, there is no trace of a gap.
To better understand how gaps arise, we need to look at which numbers fall into which bands. For small values up to about 300, the Sloan gap is less pronounced. About 18% of all numbers between 300 and 10,000 belong to the “interesting” band and the remaining 82% to the “uninteresting” values. After all, the interesting band contains about 95.2% of all square numbers and 99.7% of prime numbers, and 39% of numbers with many prime factors. These three classes already occupy nearly 88% of the interesting band.The remaining values can be 1111 or Equation 2n +1 and 2n – 1, respectively.
According to information theory, numbers that deserve special attention are numbers that have a low complexity, or are easy to represent. However, as Delahaye, Gauvrit, and Zenil argue, this can lead to the Sloane gap if mathematicians find certain values more exciting than others of the same complexity. Example: 2n +1 and 2n +2 is equally complex from an information theory point of view, but only the values of the first expression are in the “interesting band”. This is because such numbers allow the study of prime numbers. As such, prime numbers appear in many different contexts.
The division into interesting numbers and boring numbers seems to be due to the decision to emphasize prime numbers. If you want to come up with a really creative answer when someone asks you what your favorite number is, name a number like 20,067, which isn’t yet in Sloan’s Encyclopedia.
This article was originally science spectrum Reproduced with permission.
