We already know how to write 2021 Takeshi Kitano-style, using nine digits (2021 = −1 + 2 × 3 + 4 × 567 × 8/9), or just five, as Alain Zalmanski suggests (2021 = (1 + 2)!! + (3!)4 + 5).
We also know that, contrary to what Leonhard Euler imagined in 1772, 2021 could have been prime: as Fabien Aoustin pointed out, it can be written as x² + x + 41 (with x = 44). However, it is not, because 2021 = 43 × 47. Thus 2021 is the difference between two squares (452 − 22), a type of number that Pierre de Fermat was particularly fond of.
We also know, as Gaël Octavia tells us, that 2021 is the sum of forty-seven (well, well…) consecutive numbers: 2021 = 20 + 21 + … + 65 + 66. With a little searching, it can also be written as a palindrome, as François Lavallou points out: in base 46, 2021 is written XX, where X = 43.
But 2021 has plenty more tricks up its sleeve! Would you like to write it non-trivially in base 10 using just one digit? Here it is:
and it works for every digit from a = 1 to a = 9.
Would you prefer to use a single exponent? We have 2021 = 122 + 162 + 422 = 22 + 212 + 262 + 302, or 2021 = 13 + 13 + 13 + 13 + 73 + 73 + 113, or even 2021 = 14 + 14 + 14 + 24 + 34 + 54 + 64.
Can you find twenty-five features and properties of this magic square proposed by Alain Zalmanski?
Would you prefer more sophisticated representations, for example using triangular numbers of the form ? Among the results, one worth noting is 2021 = T (1) + T (19) + T (60).
Would you rather use numbers from the Fibonacci sequence, defined by
F (0) = F (1) = 1, F (n) = F (n − 1) + F (n − 2) for n ≥ 2? Here again, there are plenty of possibilities, including 2021 = F (7) + F (9) + F (14) + F (17).
We owe many of these unconventional arithmetic takes on this year's number to Inder Jeet Taneja, a former university professor in Santa Catarina, Brazil, and New Delhi, India, who specializes in recreational mathematics. He has compiled and posted online a document of more than seventy pages that will keep enthusiasts of 2021 busy for a long time.
