I am on the hunt for prime squares which decimal digits build up squares.
The smallest I have found is:
$$7^2 = 49 \cases{4=2^2\\ 9=3^2}$$
Another one is $$191^2 = 36481 \cases{36=6^2\\4=2^2\\81=9^2}$$ and $$223^2 = 49729 \cases{49=7^2\\729=27^2}$$
So we can conclude that there do exist at least a few.
Now to the question, how many of these can we find?
Can we prove there can only be finite?
Or can we estimate some distribution?
To make a more strict formulation:
Let us assume the prime square has $n$ decimal digits
neighboring digits can be grouped together in a non-overlapping way
each group consisting of at least $1$ digit and at most $n-1$ digits