Number enthusiasts may be looking to the new year with a touch of melancholy. Another perfect square like 2025 (452 = 2,025) won’t occur again until 2116 (462 = 2,116). The year 2027 will be a prime number. By comparison, our current year, 2026, seems almost boring. But that’s a misconception.
The On-line Encyclopedia of Integer Sequences (OEIS), a kind of Wikipedia for numbers, contains more than 200 entries for “2026.” This means that the number appears in more than 200 number sequences. Some of these entries are quite obscure for nonexperts, such as one that relies on an understanding of a five-cell von Neumann neighborhood. Fortunately the OEIS reveals plenty of more accessible and entertaining mathematical quirks linked to the number 2,026.
Among other things, 2,026 belongs to the group of almost prime numbers because it only has 1, 2, 1,013, and 2,026 as divisors—narrowly missing the chance to be a prime number. It can also be used to generate a prime number. It’s part of a sequence that collects prime numbers of the form 50…077 . Thus, 577, 5077, 50077 and 5000077 are prime numbers. A prime number in this sequence is 5 × 102026 + 77—that is, 5 followed by 2,026 zeros plus 77.
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The Magnetic Towers of Hanoi
The number 2,026 is related to an extension of the classic mathematical game the Tower of Hanoi. In the original game, there are three poles with disks of different sizes, each with a hole at its center. At the start of the game, the disks are placed on the first pole from largest to smallest, starting with the widest at the bottom. The goal is to move all the disks to the last pole with them arranged in the same order—but each disk may only be placed on a larger disk, never on a smaller one. It can be proven that solving the game with n disks requires at least 2n – 1 moves.
Many variations of this puzzle exist. In one particularly popular version, the disks are magnetic, with the top side representing the north pole and the bottom side representing the south pole. When you move a disk from one rod to another, you reverse its orientation: the north pole is then at the bottom. And because like poles repel each other, you now have to ensure not only that a disk can be placed on top of another one based on its size but also that the orientations of the disks are correct. This makes solving the puzzle considerably more difficult.
As it turns out, when starting with eight disks, you need at least 2,026 moves to solve the magnetic version of the Towers of Hanoi. (By contrast, if you start with three disks, you can solve the puzzle in just 11 moves.)
An Unlucky Year?
Superstitious readers should skip this section. The year 2026 could prove unlucky—at least for people who pay attention to the frequency of Friday the 13th.
Every calendar year has at least one month in which the 13th falls on a Friday—but never more than three. And 2026 is one of those years in which there is a Friday the 13th in three months: February, March and November.
This frequency last occurred in 2015. If you don’t remember all the terrible things that happened in that year, Wikipedia offers a list of natural disasters that occurred. Let’s hope the list for 2026 is shorter.
A Cheerful Number
To end on a positive note, it should be mentioned that 2026 is a so-called happy number, as popularized by British mathematician Reginald Allenby. Although there are infinitely many happy numbers there are also infinitely many sad ones. To find out which category a number belongs to, you first have to square the individual digits of the number and then add them together. That calculation for 2,026 is 2² + 0² + 2² + 6² = 44.
Then you repeat this calculation with the result 4² + 4² = 32, perform the calculation again with that result and then repeat the calculation two more times. This gives you the results 13, 10 and 1. The number 1 is the end point of the calculation, and it characterizes a cheerful number: any number that eventually ends at 1, according to the aforementioned calculation, is called happy.
Sad numbers, such as 37, on the other hand, have a different fate. For example, the calculation for 37 is 3² + 7² = 58. Continuing the calculation from this result, you arrive at 89, 145, 42, 20, 4, 16 and finally land back at 37. Sad numbers are thus trapped in a loop from which there is no escape. The only way for this calculation to end is if it eventually reach the number 1.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
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