Has anybody ever seen mentions of the following conjectures, very similar in appearance to Goldbach's or Lemoine's?
Lemoine’s conjecture is ”All odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime”
We know that it can be can be rewritten:
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 greater than 2 millions.
Additional conjecture 1: O = 2n + 1 = p - 2q always has a solution in distinct odd primes p and q for o >= 5
Additional conjecture 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 >= 1 (For every odd positive integer O there exists two distinct odd elements from PRIMES&1 p0 and p1 such that p1 is the average of O and p0 )
In addition, the Lemoine conjecture can be split into two different conjectures:
Additional conjecture 3: 2n + 1 = p + 2q always has a solution in p and q where p > q, p odd prime and q in PRIMES&1 for n > 2
Additional conjecture 4: 2n + 1 = p + 2q always has a solution in p and q where p <= q , q odd prime and p in PRIMES&1 for n > 2
Similarly the Goldbach conjecture:
E = 2n = p + q with E > 2 has a solution in primes p and q
Has a twin:
E = 2n = p – q with E > 2 has a solution in primes p and q
… but this conjecture, I believe, follows from the Goldbach conjecture and the fact that p+q which is even can always be rewritten as p-q