Problems & Puzzles: Conjectures

Conjecture 85.  Conjectures stricter that the Goldbach ones

Mark Spezzano wrote on 2-16-2020:

Conjecture 1:

     Every odd number n >= 11 can be expressed as the sum of three (higher order) primes (each of order 1, 2 and 3 respectively) in at least one way.

     That is, n can be expressed as p(i) + p(p(j)) + p(p(p(k))) for positive integers i, j, and k, where i > 1.

     Conjecture 2:

     Every even number n >= 6 can be expressed as the sum of two (higher order) primes (each of order 1 and 2, respectively) in at least one way.

     That is, n can be expressed as p(i) + p(p(j)) for positive integers i and j, where i > 1.

NOTES ON THE CONJECTURES:

If it can be proven that Conjecture 1 holds true, then if this is coupled with the results that 7 = 3 + 2 + 2 and 9 = 3 + 3 + 3 and 9 = 5 + 2 + 2 then this is enough to also prove that the Weak Goldbach Conjecture holds true.

If it can be proven that Conjecture 2 holds true, then if this is coupled with the result that 4 = 2 + 2 then this is enough to also prove that the Strong Goldbach Conjecture holds true.

Just for the record, Iíve written a computer program to check my conjectures. Conjecture 1 holds for all odd n greater than or equal to 11 and less than or equal to 15,001. (Values beyond this are unchecked) 

Conjecture 2 holds for all n greater than or equal to 6 and less than or equal to 70,000. (Values beyond this are unchecked).

Obviously, I have not found any counterexamples

Later, on my request, Mark added the following arguments in favour of his conjectures;
 

The conjectures are stricter, more constrained versions of the classical ones which is why mine have, in general, fewer solutions than their Goldbach counterparts.

 

  1. By constraining a more general result (like Goldbach) it is sometimes easier to see what is happening overall. Applying constraints is often a good way to see the important data, because it gets rid of other trivial results. I have yet to see a pattern with my conjectures, but it might enable someone else to see one.
  2. Looking at Goldbach from a new perspective or new viewpoint might help people to think about the problem from a different angle that they havenít yet considered. The given conjectures are not so obvious.
  3. I found it surprising that even with all those constraints itís still possible to find at least one solution for every given value of n. Intuitively, I would have guessed that their might be no solutions for some values of n, but this is not the case. The lowest number of solutions is at least 1. There are indeed solutions for all n (or at least it seems to be the case for the values of n I have checked so far).

Q1. Can you find a proof for these two conjectures or counterexamples?

Q2. Are these conjectures an alternative (easier/worst) approach to prove the Goldbach original ones?

Q3. Any comments related to the arguments 1-3 by Mark?

 


Dmitry Kamenetsky wrote on March 13, 2020

I checked that Conjecture 85 part 2 holds for n <= 1,000,000,000. By the way, this problem has been studied in this sequence: https://oeis.org/A237284

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