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We all know the famous Goldbach conjecture . I have found a statement says that for any even integer n >50there exists a prime p <√n such that n-p is again a prime . has any body seen this before or I am the first person to notice this conjecture which looks a little stronger than Goldbach conjecture or may be equivalent to Goldbach conjecture ... Any ideas or any thing which can be put forward to prove this ?

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    $\begingroup$ Where did you find the statement? $\endgroup$ – user236182 Dec 12 '15 at 14:28
  • $\begingroup$ I myself ... I checked a lot of even numbers and I have always found a prime less than nsquareroot for which even number-that prime is again prime $\endgroup$ – Mathslover shah Dec 12 '15 at 14:42
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It is not certain that there is any prime at all between $n$ and $n-\sqrt{n}$.
This is stronger than Legendre's conjecture, which has not been proven, and which claims there is a prime between $n^2$ and $(n+1)^2$.

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  • $\begingroup$ It's not certain but I have checked alot of even numbers which have always this property but I cannot prove it of course $\endgroup$ – Mathslover shah Dec 12 '15 at 14:46
  • $\begingroup$ OP's question only concerns even $n$. Are you sure it would imply Legendre's conjecture? $\endgroup$ – user236182 Dec 12 '15 at 14:50
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    $\begingroup$ If $n$ is even, then $n^2+2n$ is even, so OP implies there is a prime between $n^2+2n$ and $n^2+2n-\sqrt{n^2+2n}>n^2$. If $n$ is odd, OP implies there is a prime between $(n+1)^2$ and $(n+1)^2-(n+1)>n^2$. $\endgroup$ – Empy2 Dec 12 '15 at 14:57
  • $\begingroup$ Do you think what I claim implies legendre s conjecture also ? $\endgroup$ – Mathslover shah Dec 12 '15 at 15:37
  • $\begingroup$ @Mathslovershah Yes, it implies Legendre's conjecture. The comment above shows how. $\endgroup$ – user236182 Dec 12 '15 at 15:45

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