Prove or Disprove: There are infinitely many integers $n$ such that the three integers $n$, $n+2$,$n+4$ are all prime. (Suggestion: Try some sample values of $n$ and look for a pattern.)
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2At least one of them is multiple of 3. – Oct 26 '15 at 01:17
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Did you try the suggestion? It was a useful one. – rogerl Oct 26 '15 at 01:24
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I tried the suggestion. I think I was having trouble deciphering what the statement meant. ( I couldn't put it into if-then form ) – klorzan Oct 26 '15 at 01:44
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In fact, there is only one value of $n$ such that $n, n+2$, and $n+4$ are all prime, and that is $n=3$. You have $3,5,7$ are all prime.
For all other values of $n$, you should notice that exactly one of them is divisible by three. This needs to be proven however.
Break into cases:
- $n$ is a multiple of three. I.e. $n=3k$ for some integer $k$.
- $n$ is one more than a multiple of three. I.e. $n=3k+1$ for some integer $k$.
- $n$ is two more than a multiple of three. I.e. $n=3k+2$ for some integer $k$.
Now, look at what each case would imply about $n, n+2, n+4$ and their divisibility by three.
JMoravitz
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