A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, but my perception of the entire subject has. I see graphs more and more like natural numbers for which arithmetic questions are interesting. Important previous steps were to extend the natural numbers (graphs) to integers (signed graphs) then localize to get rational numbers. If done right, we get to a Banach algebra playing the role of the real numbers. In the case when only one graph is attached we get a Wiener algebra. This is an important step as we get to a structure which is no more an integral domain but still allows an effective functional calculus which is beyond analytic functional calculus. There are for example pairs of graphs A,B which are non-zero such that AB=0. Such a thing is impossible in an analytic functional calculus. Given a Banach algebra which is an integral domain (like the real numbers) and if f,g are analytic functions, then f(G) g(G) =0 implies f(G)=0 or g(G)=0. As the Wiener algebra is a Banach algebra containing smooth funtions, there is a big change.There are also quite a few little more remarks like that it is quite easy to see just by counting that most graphs are actually multiplicative primes and that we can use multiplicative linear functionals like values of the Poincare polynomial, Euler polynomial or zeta function values to get valuable information about the primes. Seeing things as part of arithmetic reduces complexity. All the rather complicated looking Fock space calculus we have in second quantizations is built in here automatically in arithmetic. The reason is that on a connection level (not incidence level) we have a tensor algebra representation. This is very, very useful. The connection graphs are under natural conditions like after one Barycentric refinement always homotopic to the Barycentric refinement of the Cartesian product so that we can see the Shannon arithmetic really doing the right thing also geometrically. That Kuenneth holds requires classically to look at CW complexes or a chain homotopy. We see Kuenneth now very concretely: given a cohomology class on two graphs, we can explicitly give the cup product representation in the product. And this works very generally for any pair of graphs.
Since most graphs are prime, a Dirichlet theorem on arithmetic sequences should be easier If A,G are two graphs with no common denominator, are there always infinitely many graphs of the form A+n G which are prime/ Most likely as for any situation in which for some multiplicative linear functional we can reduce it to the standard Dirichlet theorem, the statement applies. Unlike with the usual arithmetic, where the primes thin out, we have in the extended arithmetic of graphs most numbers are prime. The Dirichlet property might hold just for very simple reasons.
[Added June 21, 2021]: Here is an other problem which is beyond current mathematics for integers but very easy for positive dimensional graphs: Landau’s problem which asks whether there are infinitely many primes of the form . This is an open problem. But if m is a graph of positive dimension, then is always prime! The reason is that one factor necessarily has to have a zero dimensional component 1 as the product of two connected graphs is connected. That allows reduction to reduce to a smaller factor . Eventually, we need $m$ to be a rational integer (meaning zero dimension). But that means that has to be divisible by an integer. This is not possible. If A+B is divisible by an integer and A,B have no common additive prime factor (connected graph) then both A,B are divisible by that integer.]