Show there is no prime in a range of numbers How do I show that except for 5039, there is no prime between 5033 and 5047. I just need a nudge in the right direction, no idea how to start the problem :(
 A: It is also good to remember that $7! = 5040$. Hence, among the integers from $[7! - 7, 7!+7]$ the only ones that could be prime are $7! \pm 1$. But $7! + 1 = 5041 = 71^2$.
Hence, the only number that can be prime is $7!-1$.
A: Recall that a number is divisible by $3$ if and only if the sum of the digits is divisible by $3$.
If you write down each number between those two, that is $5034, 5035, \dots , 5046$ you'll notice that each one is either even, divisible by $3$ or divisible by $5$ except 5039 and 5041.  Remembering our table of squares, $71^2=5041$, so that completes proof.
A: Knowing that a composite number will always be a product of primes, let's apply some tests and see the numbers that are factors of each. Note that I may not mention all the factors since one known factor can disprove everything.
Let's begin by scratching out even numbers from the list. Now, for the odd ones.


*

*$5033:7$

*$5 0 35: 5$

*$5037:3$

*$5041:71$

*$5043:3$

*$5045 : 5$

*$5047: 7$


So, the only prime number now is $5039$ which is good enough to complete the proof.
