2021 x 1202 = 2429242, a palindrome.
That is, when you take 2021 and multiply it by its digit-reversal, you get a palindrome.
This is rare – you (if you are young) will see it again in 2101, 2102, 2111, 2201, and then not until five-digit years. It follows from the digits in 2021 being small – according to the Encyclopedia of Integer Sequences, this is a property of integers not ending in 0 with sum of squares of digits < 10.
In general we can view an integer as a polynomial – in the case of 2021, – evaluated at . Call this polynomial . Then its coefficient-reversal is where is the degree of the polynomial . For eample, if then we get the reversal Then we can show that is its own coefficient-reversal. It has degree . Upon substituting for and multiply by we get
which is itself.
Now if the coefficients of are all less than 10, we can interpret this as a fact about integers. The middle coefficient of is just the sum of the squares of the coefficients of – for example,
with middle coefficient .
For the proof that the sum of the squares is the largest coefficients, wave your hands and say “Cauchy-Schwarz”, then look at Proposition 10 of On Polynomial Pairs of Integers by Martianus Frederic Ezerman, Bertrand Meyer, and Patrick Sole.
Some other interesting properties of the number 2021: it’s a product of two consecutive primes and a value of Euler’s prime-generating polynomial. These don’t contradict each other – the polynomial is prime when evaluated at 0, 1, 2, …, 39, and .