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.

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