Who we are
Arithmetic Connectome is an independent research project founded and led by Eduardo Vega. It began with an interest in cryptography and RSA: how the structure of prime numbers can make a calculation easy in one direction and difficult to reverse.
That curiosity grew into an interest in how mathematical structure can help us solve problems—from optimizing a computation to understanding patterns in numbers. A recurring question emerged: how can understanding the relationships within a problem change the way we approach it?
Today, the project brings that question to arithmetic through computation, visual exploration, written arguments, and formal proof. AI tools help develop ideas, implement experiments, and challenge reasoning. We remain responsible for the claims we share, and welcome independent scrutiny of the work.
The path into the research
Questions we have explored
The Robin manuscript grew out of a broader investigation of arithmetic relationships. Earlier work asked how numbers can be assembled, which primes appear when they are added, and what their connections reveal. Some studies produced checked results; others showed us where an approach stops working.
01 / REPRESENTATION & DIVISIBILITY
Building integers without divisibility
In a case study of Erdős problem #123, we asked whether every sufficiently large integer can be written as a sum of distinct numbers of the form 3a5b16c, with no chosen term dividing another.
The project’s recorded Lean verification establishes this for every integer above 1,607. This is a single-triple formalization of an existing method, retained as a case study after our source review identified a general solution by others. It is not a claim of a new solution or an optimal threshold.
The original question · Erdős #123 ↗
02 / ADDITION & PRIME FACTORS
Which primes appear in pairwise sums?
Erdős problem #126 led us to sets of integers whose pairwise sums involve unusually few distinct primes. We compared structured families, residue patterns and exact finite search certificates, including five- and six-element sets in the universe through 512.
These investigations map finite configurations. A completed finite-certificate example now checks the exact prime support of one six-element witness. These finite results do not prove the original asymptotic statement about sets of increasing size.
The original question · Erdős #126 ↗
03 / GRAPHS & CYCLES
Looking for cycles of power-of-two length
For Erdős problem #64, we built bounded graph searches and independently checkable cycle certificates. The work tested small cubic graphs—graphs with three edges at each vertex—and the reliability of the search itself.
No general theorem followed from those finite checks. The study helped sharpen the distinction between finding a certificate and proving that one must always exist.
The original question · Erdős #64 ↗
04 / NUMERAL BASES & GEOMETRY
Base 60, reciprocal pairs and right triangles
Plimpton-inspired experiments connected terminating expansions, prime-factor structure and rational right triangles. We tested whether those relationships offered practical advantages for storing numbers or looking up triangle shapes.
The storage and lookup methods did not meet the experiments’ declared success criteria. The observed graph structure was explained by the construction. The consolidated E1–E3 case study retains those identities, negative findings, and reproducible experiment records. Read the case-study summary →
05 / PRIME PATTERNS & COMPUTATIONAL CONTROLS
When a striking pattern does not survive comparison
Prime-corridor experiments tracked orderings among prime pairs, including a twin-prime diagnostic through ten million. Comparisons with randomized controls and other fixed prime gaps found no unusual ordering under the registered tests.
The experiments established no limiting bias. They gave us a useful discipline: a compelling picture needs a comparison that could show it is ordinary.
We also built certificate-validation controls around the Beal equation. That work tested the checking machinery; it was not an exhaustive search or a proof attempt that established a new result. These summaries describe our recorded work, rather than a catalogue of current conjecture statuses.
These studies shaped the current research, including the approaches we chose to set aside.
