A few days ago I wrote the following post Two new conjectures related to Lemoine's and Goldbach's in which conjecture #2 was:
For every odd positive integer O there exists two distinct odd primes (from primes and 1) p0 and p1 such that p1 is the average of O and p0. I verified that the conjecture seems valid with both added constraints O < p0 < p1 or O > p1 > p0. Let's call the first case conjecture 2a and the second case conjecture 2B.
Which leads me to the following derived conjectures about triplets of odd primes...
Assume triplets [p0,p1,p2] where p0 < p1 < p2 and p1 = (p0 + p2)/2 (p1 is the average of p0 and p2) with: p0 odd prime or 1 , p1 and p2 odd primes
1) For any p0 (1 or odd prime) we can find a p1 and a p2 satisfying the above conditions. (This follows from my conjecture 2A).
2) For any p1 (odd prime) we can find a p0 and a p2 satisfying the above conditions. (This follows from Goldbach's conjecture!)
3) For any p2 (odd prime) we can find a p0 and a p2 satisfying the above conditions. (This follows from conjecture 2B).
These conjectures are of course weaker than the conjectures they derive from but I find the set of 3 pretty cool. (The link with the Goldbach conjecture is also interesting.)
Feedback? Have you seen this anywhere before?