Why Some Numbers Feel Special Across Cultures

In the paper's words

Cross‑culturally recurring “sacred” integers such as 12, 60, 108, 7 and 144 appear in myth, architecture and ritual despite disparate historical contexts. Existing explanations appeal to cultural accident or utilitarian convenience, leaving a formal, base‑invariant account absent. We fill this gap by defining a fitness functional F(n) on the integer n that combines (i) the divisor count d(n), (ii) the radical rad(n), (iii) the largest prime divisor s(n), and (iv) the depth h(n) of n in the ultrametric tree of divisibility.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Enumerate integers $n$ from $1$ to $N$, where $N$ is a chosen upper bound (here $N=120$ to include all candidate sacred numbers).

   from 3.3 Computational Procedure

2. When the bound is extended to $n\le 200$, a projection (based on the same functional) suggests that the same three numbers (144, 108, 12) would remain the top three, with $F(60)=-3.912022$; however, this extension has not been enumerated, and the maximum would remain at $n=144$ under the assumption that no $n\le200$ attains $F(n)\gt -1.974080$.

   from 4.4 Comparison with Nearby Candidates

3. Upper bound $N=120$ (or $200$) Larger $N$ might introduce numbers with higher $d(n)$ that overcome the smoothness penalty.

   from 6.1 Limitations

4. Computational: Efficient prime factorisation for numbers up to $10^{6}$ is required to explore larger $N$.

   from (d) Engineering Constraints

5. Assuming a single researcher dedicates 40 h to implement the simulation (hourly rate $ 50) and a modest cloud compute instance costs $ 0.10 h, the total projected cost is: $$ \text{Labor}=40\text{ h}\times\$50/\text{h}= \$2000, \quad \text{Compute}=40\text{ h}\times\$0.10/\text{h}= \$4, $$ giving an overall $ 2004 budget (projection, based on assumptions above).

   from (e) Projected Cost Bounds (Projection)

How it is done

1. Using the explicit formulas $d(n)=\prod_{p^{e}\parallel n}(e+1)$, $\operatorname{rad}(n)=\prod_{p\mid n}p$, $s(n)=\max\{p\mid p\text{ prime},p\mid n\}$ and $h(n)=\sum_{p^{e}\parallel n}e$, we compute $F(n)=\log d(n)-\log\operatorname{rad}(n)-\log s(n)-\log h(n)$.

   from Abstract

2. Separately, [4] treats landscape dynamics, interbasin kinetics and ultrametric diffusion; we use it only as topical motivation for reading $F(n)$ as an energy on a hierarchical landscape, and invoke no dynamical result of [4].

   from 2. Background and Related Work

3. For any positive integer $n$ we define the valuation vector $$ v(n)=\bigl(\operatorname{rad}(n),\;d(n),\;s(n),\;h(n)\bigr) $$ where: $\operatorname{rad}(n)=\displaystyle\prod_{p\mid n}p$ is the radical , i.e. the product of distinct prime factors of $n$. $d(n)=\displaystyle\prod_{p^{e}\parallel n}(e+1)$ is the divisor count (standard number‑theoretic function). $s(n)=\displaystyle\max\{p\mid p\text{ prime},\,p\mid n\}$ is the smoothness index , i.e. the largest prime divisor. $h(n)=\displaystyle\sum_{p^{e}\parallel n}e$ is the tree depth , the total exponent sum in the prime factorisation of $n$.

   from 3.1 Valuation Vector

4. We construct a scalar fitness functional $F:\mathbb{N}\to\mathbb{R}$ that rewards high divisor density, low radical (i.e. many small primes), low smoothness (i.e. smooth numbers), and shallow tree depth (i.e. compact factorisation).

   from 3.2 Fitness Functional

What would falsify it

1. If a comprehensive ethnographic database covering at least 500 cultures shows that a number outside $\{12,60,108,7,144\}$ dominates as the most frequently cited sacred integer, then the hypothesis that the ultrametric fitness functional predicts cultural salience is falsified.

   from 6.2 Falsification Criterion

2. The result predicts that agents whose decision‑making weights both magnitude and divisibility will preferentially converge on these numbers.

   from Abstract

3. If the prediction holds, a unified non‑Archimedean explanation of numerical symbolism emerges, linking $p$‑adic geometry to cognitive preference.

   from Abstract

Where the claim stops

1. Assumption Potential Sensitivity Choice of logarithmic aggregation (Equation 1) Different weightings could reorder the top candidates.

   from 6.1 Limitations

2. Use of natural logarithm Base change scales all terms uniformly and does not affect ordering, but extreme values could be sensitive.

   from 6.1 Limitations

3. Future work will test the prediction through agent‑based simulations and large‑scale ethnographic correlation, thereby moving the theory from TRL 2 toward empirical validation.

   from 8. Conclusion

Where to read

1. Abstract241 words · 1 min
2. 1. Introduction307 words · 1 min
3. 2. Background and Related Work478 words · 2 min
4. 3. Methods254 words · 1 min
5. 4. Analysis335 words · 1 min
6. 5. Results152 words · 1 min
7. 6. Discussion204 words · 1 min
8. 7. Practical Implications and Engineering Pathways267 words · 1 min
9. 8. Conclusion98 words · 1 min
10. Provenance and AI Involvement225 words · 1 min