2021

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, $2x^3 + 2x + 1$ – evaluated at $x = 10$ . Call this polynomial $f(x)$ . Then its coefficient-reversal is $x^{deg(f(x))} f(1/x)$ where $deg(f(x))$ is the degree of the polynomial $f(x)$ . For eample, if $f(x) = 2x^3 + 2x + 1$ then we get the reversal $x^3 (2/x^3 + 2/x + 1) = 2 + 2x^2 + x^3.$ Then we can show that $g(x) = latex f(x) f(1/x) x^{deg(f(x))}$ is its own coefficient-reversal. It has degree $deg g(x) = 2 deg f(x)$ . Upon substituting $1/x$ for $x$ and multiply by $x^{2 deg f(x)}$ we get

$f(1/x) f(x) (1/x)^{deg f(x)} x^{2 deg f(x)} = f(x) f(1/x) x^{deg f(x)}$ $

which is $g(x)$ itself.

Now if the coefficients of $g(x)$ are all less than 10, we can interpret this as a fact about integers. The middle coefficient of $g(x)$ is just the sum of the squares of the coefficients of $f(x)$ – for example,

$(x^3 + 2x + 1) (x^3 + 2x^2 + 2) = 2x^6 + 4x^5 + 2x^4 + 9x^3 + 2x^2 + 4x + 2$

with middle coefficient $2^2 + 2^2 + 1^2 = 9$ .

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 $n^2 + n + 41$ is prime when evaluated at 0, 1, 2, …, 39, and $2021 = 44^2 + 44 + 41$ .