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

I know $1000!$ is not a prime number as any number $1000$ or less is a divisor, but how would I know if $1000!+1$ is prime? Any hints?

Also, use the above question to prove that you can find $n$ consecutive composite numbers.

• You have only commented on determining whether or not $1000!+1$ is prime. What about the others in the range, do you have any thoughts on them? – Michael Albanese Dec 19 '13 at 3:17
• $n!+r$ is divisible by $r$ for $2\le r\le n$ – lab bhattacharjee Dec 19 '13 at 3:18
• the question explicitly says inclusive – cats Dec 19 '13 at 3:19
• Your question about $1000!+1$ may be difficult. – André Nicolas Dec 19 '13 at 3:19
• $n!+1$ is this sequence; I concur with @AndréNicolas that knowing if it's prime is not easy. – vadim123 Dec 19 '13 at 3:22

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.

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

• $k\ge0$ and $\ne1$ right? – lab bhattacharjee Dec 19 '13 at 3:18
• It's still true for $k=1$, however it doesn't exclude primality. – vadim123 Dec 19 '13 at 3:20
• That's a good hint when $k\ge 2$. At $k=1$ it's a big number, not clear whether prime or not. – coffeemath Dec 19 '13 at 3:20

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.

• Please explain how to do this simple check. – vadim123 Dec 19 '13 at 14:33
• It was a check using software. I used this command in pari/gp: N=1000!+1;forprime (p=1003,1000000,if (N%p==0,print(p))). Just to see if there were any small prime factors, and I was lucky. – Old John Dec 19 '13 at 14:37

I cannot give a proof, but the OEIS entry on factorial primes claims that $1000!+1$ is not prime, as 1000 isn't listed. This is likely a proof-by-computer, though.

As for the rest of the numbers in that range, everyone else has already mentioned that $k\mid n!+k$ for $1\leq k\leq n$.

You can use wolfram alpha to check. Not really elegant but effective :-)

http://www.wolframalpha.com/input/?i=Is+1000%21%2B1+prime%3F