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.