# For any integer $n$ greater than $1$,how many prime numbers are there greater than $n!+1$ and less than $n!+n$?

For any integer $n$ greater than $1$,how many prime numbers are there greater than $n!+1$ and less than $n!+n$ ?

By trying different values of $n$ for $n=2,3,4,5,6$ I get a feeling that the number of primes in the interval is $0$,of course this might be wrong as $n$ can be any number.

I've not been able to do much progress on this problem...one idea I had was to prove the above conjecture by proving it by picking up first smaller intervals and prove it works for those intervals.

My feeling is also that Wilson's Theorem might be applicable here.

## 2 Answers

Hint: Try to find a divisor of $n! + k$, $1 < k \le n$, within the range $[1,n]$.

• How would I go about finding a factor of $n!+k$ ?I don't know any other closed form of that expression... – Mr. Y Jan 23 '16 at 8:56
• Try for some small $n$ and $k$. I think you will see a pattern. – gerw Jan 23 '16 at 9:55
• Maybe I see it.If we start from $n!+n$ this can't be a prime since $n!+n=n((n-1)!+1)$ and $n>1$,similarly by going backwards $n!+n-1=n(n-1)!+(n-1)=(n-1)((n-2)!+(n+1))$ which can't be a prime.In general $n!+n-k$ can't be a prime as it can be factored as the product of two numbers greater than $1$ since $n-k < n$ and so $n!$ contains it as factor .Last option $n!+1$ but $n!+1=n!+n-(n-1)$ and if we let $n-1=k$ we have the same thesis as above. – Mr. Y Jan 23 '16 at 10:47
• Above I forgot to mention that $1<k \le n$. – Mr. Y Jan 23 '16 at 10:54
• Yes, $k$ is a divisor of $n! + k$ for all $1 < k \le n$. But $n! + 1$ might be prime. – gerw Jan 23 '16 at 11:59

Hint: Let $a = n! + 7$. Does $7 \mid n!$? Does $7|7$? Does $7 \mid n! + 7$? Is $n! + 7$ prime?

If it isn't what number greater than $n! + 1$ and less than or equal to $n! + n$ might be?

Can you factor anything out of the number $n! + k$?

• $n!+1$ might not be prime. – gerw Jan 23 '16 at 9:56
• $n! + 1$ might not be prime but it's the only one could be prime. The question was "greater than n! +1" , I ambiguously wrote "between" when I should be have written "greater than". I've fixed that. (On the other hand, the original question was "less than" $n! + n$ but obviously $n! + n$ isn't prime. – fleablood Jan 23 '16 at 17:19