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 :(
Show there is no prime in a range of numbers
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2@Arthur: Eric’s argument takes care of everything but $5041$. When it doesn’t succumb to the easy tests for $2,3,5$, and $11$, the fact that $50$ is just a little over $7^2$ and the number ends in $1$ could suggest trying $71^2$. Even if you think that you’ll have to test everything the hard way, you’ll want to know how far you need to go, so you’ll want $\sqrt{5041}$ anyway. – 2011-11-09
3 Answers
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 $7!-1$.
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.
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0You didn't mention _divisible by 7_ BTW. 5033 and 5047 are both divisible by 7. =) – 2012-12-22
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.