Can you determine small primes from larger primes? Suppose you are given the primes in the range $[n,n^2]$. Is there a known way to effectively reconstruct the primes less than $n$? Ideally, something that takes less calculation than figuring them out directly from sieving, as that would sort of defeat the purpose.

Heuristically, you can do pretty well by taking $n \pmod{p}$ where $n$ is composite and $p$ steps through your prime list, and then seeing what comes up most often, but I'm hoping for something better than that.
This is very slow compared to e.g. the sieve method. For example with $n=1000$ there are $\pi(10^6)-\pi(10^3) = 78330$ primes in $[n,n^2]$ but only $168$ primes less than $1000$.
So you would do $78330$ modulo operation with your method. For larger $n$ your method would even be worse.
This is very slow compared to e.g. the sieve method. For example with $n=1000$ there are $\pi(10^6)-\pi(10^3) = 78330$ primes in $[n,n^2]$ but only $168$ primes less than $1000$.
So you would do $78330$ modulo operation with your method. For larger $n$ your method would even be worse.