Take the numbers of the current and next year $(2020,2021)$
I noticed that $ 2020 = 2* 1010 $ and if we take the square of theses divisors added by $2021$ we get the squares $45$ and $1011$.
Or also $$ 2021 = 45^2-2^2 $$ and $$ 2021 = 1011^2 - 1010^2 $$
What is the next year, if there is one, where it will happen that for both squares of the divisor of the old year plus the new year are square?
On
Let $c$ be any odd integer $3$ or greater, let $b=(c^2-5)/2$, and consider the numbers $2b$ and $2b+1$. We have $2^2+(2b+1)=c^2$, and $b^2+(2b+1)=(b+1)^2$.
The original example is $c=45$, $b=1010$ giving $2b=2020$, $2b+1=2021$.
Axel's example is $c=47$, $b=1102$ giving $2b=2204$, $2b+1=2205$.
The last previous year pair would have come from $c=43$, $b=922$, $2b=1844$, $2b+1=1845$.
Clearly, one gets an infinity of examples from these formulas.
All of these examples come from using $2$ as one of the divisors of the first year of the pair of consecutive years. It's a harder problem to find examples that don't use $2$ as a divisor.
Well I wrote a program, and the answer is $2204 = 2\times1102$.
Most of the times (I have tested for $n \leq 50000$ only) the only divisor is $2$, so you end up with $n = 2 \times (n/2)$. There are some quite remarkable numbers though :$144=12\times 12$ $455=13\times35 $ , $4900 = 70 \times 70$, $5719=43\times 133$, $26676=78\times 342$, $41040=180\times 228$.
See you in $2204$.