Numerical Experiments — Goldbach Research

Empirical observation
Everything on this page is an empirical observation by the author, not a theorem. Spot checks are not exhaustive searches and prove nothing. The numerical material is presented as a diagnostic instrument for hypotheses, not as part of any proof.

External anchor: complete verification up to 4·10¹⁸

The binary Goldbach conjecture — the assertion that every even $N \geq 4$ is a sum of two primes — has stood unresolved since 1742. The only form of rigorous knowledge about it “at large” is exhaustive verification.

The largest verified range is due to Oliveira e Silva, Herzog, and Pardi: in a paper published in Mathematics of Computation (vol. 83, 2014), the conjecture is verified for all even numbers up to $4 \cdot 10^{18}$ . Published article: doi:10.1090/S0025-5718-2013-02787-1; the project page with the data, formerly at sweet.ua.pt, is preserved in the Internet Archive.

The present project relies on this verification as an external fact: in the range $N \leq 4 \cdot 10^{18}$ the question is settled by computation, and the author’s own calculations there serve only to calibrate code. The author’s experiments begin where exhaustive search is no longer possible.

Own spot checks

Write $r(N)$ for the number of representations of an even $N$ as a sum of two primes:

r(N) = \#\{(p, q) : p + q = N,\ p \le q,\ p, q \text{ prime}\}

The author has carried out two families of spot checks of the condition $r(N) > 0$ :

Sample	Range	Outcome
121,600+ random even $N$	$[\,5 \cdot 10^{18},\ 10^{500}\,]$	a representation found for every point tested: $r(N) > 0$
Powers of ten, $N = 10^{k}$	$k = 20, \ldots, 1000$	$r(N) > 0$ confirmed at every point

Empirical observation
These are checks of individual points, not searches over intervals: unbridgeable gaps remain between the tested numbers. Such tests have statistical meaning — they show that the observed picture does not contradict the conjecture — but no probative force. No finite sample can replace a proof for all even numbers.

Empirical regularities

Four regularities reproduce stably across the computations. Each carries the status Empirical observation — these are observations over finite data, not proved statements.

#	Regularity	Status and caveat
1	Smallest prime in a representation: $p_{\min}(N) \approx 0.62 \log N \cdot \log\log N$ .	Empirical observation
2	Minimum relative to expectation: $\min r(N)\,/\,\text{expectation} \geq 1.11$ for $N \leq 200\,000$ .	Empirical observation
3	Spread: $\sigma(N) \propto (\log N)^{-3.9}$ (fit with $R^{2} = 0.9997$ ).	Empirical observation
4	Decorrelation: $C_{h} / C_{0} \approx 0$ for $h \geq 1$ .	Empirical observation cancellation mechanism open

The last item merits a separate comment. The observed vanishing of the autocorrelations at every nonzero shift $h \geq 1$ suggests that the contributions of major and minor arcs cancel one another (major–minor cancellation). The mechanism of this cancellation is not understood and is recorded in the project as an Open problem: to explain it would be to advance the understanding of why the heuristic agrees so well with the data.

Numerical support for conjecture CWC₃

Conjecture CWC₃ (its statement and status are given on the Publications page) predicts sublinear growth of a certain sum $\Sigma(X)$ . Computations have been carried out up to $X = 5000$ :

Quantity	Value
Observed value	$\|\Sigma(5000)\| = 11\,491$
Sublinear threshold	$2\,935\,239$
Ratio	$11\,491\,/\,2\,935\,239 \approx 0.004$

numerical support, not a proof
The observed value is about 0.4% of the sublinear threshold — consistent with conjecture CWC₃ and supportive of it. However, CWC₃ remains a Conjecture: a finite check up to $X = 5000$ says nothing about the behaviour of the sum as $X \to \infty$ .

Spectral audit of the defect

Define the defect — the deviation of the observed number of representations from the heuristic prediction involving the singular series $\mathfrak{S}(N)$ :

\Delta(N) \;=\; r(N) \;-\; \mathfrak{S}(N)\,\frac{N}{\log^{2} N}

The spectral audit decomposes the defect into modes associated with the nontrivial zeros of the Riemann zeta function $\zeta(s)$ . The result:

Component	Share of the variance of $\Delta(N)$
Low modes: the first 100 nontrivial zeros of $\zeta(s)$ ( $\gamma \leq 1419$ )	at most 13%
High-frequency tail of the spectrum, not captured by the low modes	approximately 87%

Empirical observation
These are the author’s internal experiments; the methodology has not been published. The conclusions should be regarded as a working picture for internal use, not as an established fact.

The role of numerics in the project

The computations collected here serve two auxiliary functions:

Diagnostics. Before months are invested in an analytic proof, the idea is tested against the data: if the heuristic predicts one thing and the numbers show another, the error is to be sought in the idea, not in the arithmetic.
Killing false directions. A quick numerical test can dispose in a single evening of a hypothesis whose attempted proof would have consumed years. This is the most economical function of experiment.

What numerical experiments never do in this project is serve as a base for proof: no finite computation can confirm a statement about all even numbers. The criteria of rigour and the hierarchy of statuses are described on the Methodology page.