Lemoine's conjecture can be written:

$2n + 1 = p + 2q$ always has a solution in primes $p$ and $q$ (not necessarily distinct) for $n > 2$.

It appears that the following very similar conjectures (also applying to the relations between an odd number, a prime and a semi-prime) can also be verified for values of o greater than 2 millions.

  1. $O = 2n + 1 = p - 2q$ always has a solution in distinct odd primes $p$ and $q$ for $O \geq 5$
  2. $O = 2n + 1 = 2q - p$ always has a solution in $p$ and $q$ with $q < p$, $p$ being an odd prime and $q$ being 1 or an odd prime and $O \geq 1$

The second can also be written as:

For every odd positive integer $O$ there exists two distinct odd primes (from primes and 1) $p_0$ and $p_1$ such that $p_1$ is the average of $O$ and $p_0$.

which makes it very similar to one of the possible formulations of Goldbach's seen in Is every composite number the average of two primes?

Has anybody seen these two conjectures anywhere before?

  • $\begingroup$ I like your spacing. Your use of $O$ and $o$, not so much, $\endgroup$ – Robert Soupe Sep 17 '17 at 1:37

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