Fans of numbers may be looking at the new year with a touch of melancholy. Another perfect square like 2025 (452 = 2,025) will not 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 this is a misconception.
THE Online encyclopedia of entire sequences (OEIS), a sort 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 non-experts, such as one that relies on understanding a five-cell von Neumann neighborhood. Fortunately, the OEIS reveals many more accessible and entertaining mathematical oddities related to the number 2,026.
Among other things, 2026 belongs to the group of near-prime numbers because it only has 1, 2, 1013, and 2026 as divisors, narrowly missing the chance of being a prime number. It can also be used to generate a prime number. This is part of a sequence which collects prime numbers of the form 50…077 . So, 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 tours of Hanoi
The number 2,026 is linked to an extension of the classic math game THE Hanoi Tower. In the original game, there are three poles with different sized discs, each with a hole in its center. At the start of the game, the discs are placed on the first post from largest to smallest, starting with the widest at the bottom. The goal is to move all the disks to the last pole by placing them in the same order, but each disk can only be placed on a larger disk, never on a smaller disk. We can prove that solving the game with n disks requires at least 2n – 1 shot.
Many variations of this puzzle exist. In a particularly popular version, the discs 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 as the poles repel each other, we must now ensure not only that one disk can be placed on another depending on its size but also that the orientations of the disks are correct. This makes solving the puzzle considerably more difficult.
It turns out that when you start with eight discs, you need at least 2,026 moves to solve the magnetic version of the Towers of Hanoi. (On the other hand, if you start with three discs, you can solve the puzzle in just 11 moves.)
An unhappy year?
Superstitious readers should ignore this section. The year 2026 could prove unlucky, at least for those paying attention. Friday the 13th frequency.
Each 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 where 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 that year, Wikipedia has a list of natural disasters which happened. Hopefully the list for 2026 will be shorter.
A happy number
To end on a positive note, it is worth mentioning that 2026 is a so-called happy number, as popularized by British mathematician Reginald Allenby. Even if there are an infinite number happy numbers there are also an infinite number of sad ones. To find out which category a number belongs to, you must first square the different digits of the number and then add them together. This calculation for 2026 is 2² + 0² + 2² + 6² = 44.
Then you repeat this calculation with the result 4² + 4² = 32, perform the calculation again with this 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 happy number: any number that ends in 1, according to the aforementioned calculation, is called happy.
On the other hand, sad numbers, like 37, experience 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 at 37. The sad numbers are thus trapped in a dead-end loop. The only way for this calculation to complete is if it eventually reaches the number 1.
This article was originally published in spectrum of science and has been reproduced with permission.
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