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In how many ways can $10001$ be written as the sum of two primes?

Obviously since the 10001 is odd, one of the primes must be $2$. This leaves the second, must be prime as 9999, but it isn't, hence there are $0$ ways to write $10001$ as the sum of two primes.

Another way I would appreciate if someone can give me feedback on is:

All primes $p$ can be expressed as either $p=6k+1$ or $p=6k-1$, where $k$ is some positive integer. If I sum two distinct primes I have three possibilities: $$p_1+p_2=(6k+1)+(6m+1)=6(k+m)+2$$ or $$p_1+p_2=(6k-1)+(6m+1)=6(k+m)$$ or $$p_1+p_2=(6k-1)+(6m-1)=6(k+m)-2$$

Since $10001$ is neither even, which would satifie (1) or (3) and since it is not multiple of 6, which would satisfy (2), there exists no primes that would sum to $10001$. Is this logic correct?

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    $\begingroup$ The first approach is good. For the second one, note that $2$ and $3$ are primes but they cannot be expressed as $6k\pm 1$ for a positive integer $k$. That may be a gap you may want to fill in. $\endgroup$
    – Lord Soth
    Oct 30, 2013 at 2:22
  • $\begingroup$ Other than those small counter exceptions, the logic seems very good! $\endgroup$
    – Bennett Gardiner
    Oct 30, 2013 at 2:23
  • $\begingroup$ That statement seems to be only valid for all primes 5 and above. I won't see how 3 falls into 6k+1 or 6k-1. Moreover, all three options results in an even answer. 103 can be written as the sum of 2 primes, 2+101. But 103 falls in neither category. $\endgroup$
    – imranfat
    Oct 30, 2013 at 2:25
  • $\begingroup$ All primes EXCEPT $2$ and $3$ are of the form $6k\pm 1$. $\endgroup$
    – Michael Hardy
    Oct 30, 2013 at 2:27

2 Answers 2

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The primes 2 and 3 can't be expressed as 6k+1 or 6k-1 with k an integer, so you need to check those, but otherwise your solution is fine.

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Suppose a prime number is $p$ and $p$ is to be written as sum of other two primes $q$ and $r$. $$q+r=p\Leftrightarrow q=p-r$$ squaring both sides : $$q^2=p^2+r^2-2pr\Leftrightarrow q^2+2pr=p^2+r^2$$ Since every prime greater than $2$ is odd therefore the left hand side is odd whereas the right hand side is even(sum of two odds).therefore a prime cannot be written as sum of two primes otherthan twin primes

I was not answering whether the logic was correct or not, i just gave my own proof of it.

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    $\begingroup$ This might be more general, but it does not answer the question whether the OP's logic is correct. $\endgroup$
    – user228113
    Jun 24, 2015 at 16:41

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