I am investigating a combinatorial structure on prime numbers called an n-Level Decomposition Tree (DT$_n$). The definition is as follows:
Let $p > 17$ be a prime number. An n-level decomposition tree of $p$ is a rooted tree defined recursively:
- The root of the tree is $p$.
- Each non-leaf node decomposes into three distinct odd primes whose sum equals the value of that node.
- Child nodes are decomposed similarly, and all nodes in the tree are distinct from each other and from their parent nodes.
- The tree has depth $n$, where the root is at level 0.
The total number of nodes in an n-level tree is at most:
$$ \sum_{i=0}^{n} 3^i = \frac{3^{n+1}-1}{2}. $$
Hypothesis (DT$_n$): For any positive integer $n$, there exists a sufficiently large prime $p$ such that the maximum depth of its decomposition tree is exactly $n$.
Example:
$$ 127 = 37 + 43 + 47, \quad 37 = 7 + 13 + 17, \quad 43 = 3 + 11 + 29, \quad 47 = 5 + 19 + 23 $$
gives a tree of depth 2 with 13 distinct primes:
$$ \{127, 37, 43, 47, 7, 13, 17, 3, 11, 29, 5, 19, 23\}. $$
Question: Does there exist a general method to construct such an n-level decomposition tree for arbitrarily large $n$? Is there any known result regarding the existence or density of primes that admit an n-level decomposition tree of depth $n$?
Any references, related results, or insights into combinatorial structures of prime sums would be greatly appreciated.
