How many prime numbers are there in between $1000!+1$ and $1000!+1000$, inclusive?

Solution 1:

The only non-trivial case is $n=1000!+1$.

However, you can easily check with a computer that $2^{n-1} \not \equiv 1 \pmod n$, thus it's not a prime number (it's just an instance of Fermat primality test). If you want to try this yourself, use an efficient modular exponentiation method.

You may also have a look at FactorDB, which will give you a partial factorization: $$1000! + 1 = 6563 \cdot 1190737 \cdot 115205557790605547 \cdot C_{2541}$$ where $C_{2541}$ is a composite number with 2541 decimal digits.

Solution 2:

Hint: $k$ divides $1000!+k$, for every natural $k\le 1000$.

Solution 3:

All the integers from $1000!+2$ up to $1000!+1000$ are clearly not prime, and a simple check verifies that $6563$ is a factor of $1000!+1$, so there are no primes in the list.