As we approach the end of the year, I thought about the appearance of primes in calendar dates as a recreational problem. Consider the number formed by concatenating the digits of a date in the form YYYYMMDD. For example 31-Dec-2019 will be written as $20191231$.
I was investigating if the number YYYYMMDD is prime. I checked for the next hundred thousand years and found that each year has between a minimum of $1$ for the year $5771237$ and a maximum of $37$ primes for the year $450060$. I could not yet find a single year which did not have a prime.
Conjecture: There is at least one prime every year.
Update: The year $27789755$ is the smallest year which has no prime.
What is the smallest counter example?
Also $37$ primes occurring in the year $450060$ because it implies that the interval $(4500600001, 4500601231)$ contains at least $37$ primes. Upon checking, it turns out this interval contains $77$ primes which is quite a dense for a short interval between two large numbers.