Welcome to the twenty-first year of the second millennium. When I was a boy, 21 meant coming of age, when you got the “key of the door” and were free to go off and do your own thing. Of course, Mum and Dad were usually still around to come to the rescue if things didn’t work out first time. Well, today the United (but for how much longer?) Kingdom has set off to do its own thing, free from the thraldom of Brussels. Only time will tell if we are up for the task. If only we could pull up our island's anchor and set sail to more congenial climes, and seek out new neighbours who don't want to snaffle our fish.
The number 21 also has gambling connotations. It is the number of spots on a standard die, and the name of a card game adopted by the British as Pontoon (or Vingt-un if they wanted to sound posh). Are we still a gambling nation, are we again willing to take risks for the sake of enterprise and prosperity? Or are we lost in dreams of our “glorious past” when the sun never set on the possessions and peoples we then held in our thrall, now too tired to summon the energy, initiative and ambition to make a fresh start? We shall see. Meanwhile, let’s indulge in some trivial distractions with the number 2021 itself and briefly postpone the big challenge of facing up to a new future in an unfriendly world.
Is 2021 a prime number? To check that we only need look for prime divisors up to the square root √2021 = 44.9555… There are 14 primes less that 45 and you have to go to the wire to discover that the 14th prime 43 is the smallest prime divisor of 2021. In fact, 2021 has only two prime factors: 2021 = 43 x 47 and is not quite a perfect square. (We will have to wait four more years to find a year that is a perfect square, namely 2025 = 452 — the previous one, 442 was the year of my birth.) Anyway,, the answer to our initial question is “No”, 2021 is not a prime number.
We can represent the familiar decimal form of our current year 2021 in different bases, for example:
Base-2 (binary): 11111100101
Base-3: 2202212
Base-16 (hexadecimal): 7e5 = 7 x 162 + 14 x 16 + 5
Or we can devise other recipes, such as finding expressions that use the four operations of arithmetic and the digits 1 — 9 in order as follows:
2021 = 1 - 23 - 4 + (5 x 6 - 7) x 89
2021 = 12 × (3 × 4 + 5 + 6) × 7 + 89
2021 = 1 x (2 + 3 ) - (((4 x 56) x (7 - 8)) x 9), or
2021 = (1^23 + 45) x 6 x 7 + 89 using a fifth operation exponentiation
2021 = (9 × 8 + 7 + 6) × 5 × 4 + 321
Instead of using all the digits, we can stick to just one digit and try to find the most efficient expression (i.e. the one with fewest occurrences of the digit). Here are some examples:
2021 = ((11 - 1) - 111) x ((1 - 11) + (1 - 11)) + 1 =
2/2 + (22/2 - 2) x 222 + 22 =
333 x (3 + 3 ) + 3 x 3 x 3 - 3 - 3/3 =
(44 - 4/4) x (44 + 4 - 4/4)
or, most inefficiently, 2021 = 9/9 + 9/9 + .. (2021 terms).. + 9/9
So I will leave readers who, like me, are amused by such bagatelle with the 2021 challenge of finding the shortest single-digit expression(s) for 2021. You have the rest of the year to improve on my best short score, which is 9.
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