TITLE: Is there a prime every year if YYYYMMDD is treated as a base-$10$ number?
QUESTION [24 upvotes]: 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.

REPLY [18 votes]: It is false, as you can construct sequences of composite numbers of arbitrary length $n$, e.g.,
$$(n+1)! + 2, \ldots, (n + 1)! + (n + 1) :$$ Pick such a sequence of length at least $19999$, so that it contains a subsequence of consecutive numbers ending in $0000, \ldots, 9999$. Then, the first number in that subsequence is $10\,000 \cdot y$ for some integer $y$, and all the numbers given by date strings from the year $y$ are composite.