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Is there any counterexample of the following statement?

"Every odd number( $>5$) can be written as $2p+q$ where $p$ and $q$ are prime numbers."

Clyde Kertzer
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Mohammad Bagheri
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    It has a whiff of the Goldbach conjecture, where while no counterexample has been found, has not been proven either. – user317176 Apr 14 '22 at 01:12
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    General reference: Dickson's conjecture. Search on that to learn more about questions of this type. – Bill Dubuque Apr 14 '22 at 01:35
  • @DougM How do you know? Has a computer search been done somewhere? – Clyde Kertzer Apr 14 '22 at 02:44
  • @ClydeKertzer I know nothing about the OP's conjecture. I do know a little bit about Goldbach's conjecture (i.e. any even number can be written as the sum of two primes). – user317176 Apr 14 '22 at 04:10
  • Ah, I understand. I thought you said there was no counterexample to the OP's conjecture. – Clyde Kertzer Apr 14 '22 at 04:15
  • I am not sure whether any of the known conjectures as Goldbach's conjecture or Dixon's conjecture implies this conjecture. If so, this should be worked out in a comment before this is considered to be a duplicate. – Peter Apr 14 '22 at 08:50
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    Upto $243\ 377\ 803$, the conjecture is true : More precise , there is a prime $p\le 1483$ doing the job. – Peter Apr 14 '22 at 11:59
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    Upto $766\ 707\ 661$ , there is a prime $p\le 1609$ doing the job. – Peter Apr 14 '22 at 12:47
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    Harald Helfgott proved the weak Goldbach conjecture: if $n>3$ you can find odd primes $p$, $q$ and $r$ so that $2n+1= p+q+r$. You conjecture is much stronger than this, as you require $p=r$ (though you do also allow the case $p=r=2$ which slightly weakens it again).

    If the Goldbach conjecture is true then the weak Goldbach conjecture is a trivial consequence. Your conjecture does not seem to follow from the Goldbach conjecture and is probably equally hard.

    As even the weak Goldbach conjecture was very hard to prove, I don't think you'll get a definitive answer for some time.

     – Robin Balean Apr 14 '22 at 18:27
  • Thanks for all useful comments. @Peter, could you please tell us about the number of ways which an odd number $N$ can be written by $p$ and $q$?(Is it increasing as $N$ grows?) – Mohammad Bagheri Apr 14 '22 at 21:10

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