A Graphical View of Prime Summands

The conjecture considered in the previous post is that each prime number greater than 3 can be written either as the sum of two distinct primes or as the sum of two distinct primes plus 1.

For the prime 5, the graph is very simple

Here are some other graphs for larger primes:

As you can see, these graphs become hard to read as the number of primes increase. Nonetheless, we will present in a future blog post the python code used to generate these diagrams.

These small graphs, however, suggest another conjecture. Note that the nodes for 1,2,3 are the only source nodes, i.e, nodes of in-degree 0. Likewise, the only sink node, i.e., a node of out-degree 0, is the largest prime in the graph. All of the other intermediate primes have non-zero in- and out-degrees. Is this generally true? It is easy to see that by design none of the primes larger than 3 will be source nodes (if the conjecture is true), but can sink (or leaf) nodes appear among intermediate primes at some point for some large prime p? That is an interesting conjecture worth considering and it also doesn't seem to be equivalent to the main conjecture we have been considering. Stay tuned for more discussion!