On the Distribution of Smaller Factors in Semiprimes

Jonathan Kendall, 2026

An empirical study of the coordinate θ = log p / log n for semiprimes n = pq (p ≤ q). Exact enumeration up to N ≤ 10⁸ shows that the proportion P(N, θ₀) of semiprimes with θ ≤ θ₀ increases monotonically with N, consistent with O(1/log log N) decay toward a limiting value — the classical rate for related semiprime counts (Montgomery–Vaughan §7.4). A novel discrete activation phenomenon is documented: each prime p contributes a visible step in P(N, θ₀) when N crosses p^(1/θ₀). A refined heuristic prefactor f(θ₀) = log(1/θ₀) − θ₀ − log 2 + ½ reduces the maximum relative prediction error for δ · log log N from 4.3× (naive Mertens) to under 1.21× across θ₀ ∈ [0.10, 0.45]. Several geometric and spectral embeddings (additive phase wrapping, multiplicative phase maps, spectral fingerprint, factor-ratio spectrum) all fail to recover the structure carried by θ, consistent with a separation principle: factor asymmetry is not recoverable from any function of n alone that is multiplicative or additive in log-space.

Paper

"On the Distribution of Smaller Factors in Semiprimes: Discrete Activation Thresholds, Slow Convergence, and a Refined Heuristic" — Jonathan Kendall, 2026

📄 Read the paper (PDF) | [Submitted to Integers: Electronic Journal of Combinatorial Number Theory]

To reproduce all paper results exactly:

python3 scripts/verify_semiprime_tables.py

The PrimeHelix toolkit (CLI, visualizations, structure-scan) is a separate research tool built alongside the paper.

Requirements

• Python 3.8+, numpy, scipy, matplotlib
• ~400 MB RAM for SPF sieve to N = 1e8
• Runtime: ~8 minutes

Reproduce all results

python3 scripts/verify_semiprime_tables.py

scripts/verify_semiprime_tables.py is the canonical script (commit fc43ef0). scripts/verify_paper_tables.py is an earlier version kept for reference only.

Outputs

• results/table1.csv, table2.csv, table3.csv
• activation_theta_0_10.png
• stabilisation.png

Verified checksums (exact, no sampling)

• 210,035 semiprimes ≤ 1e6
• 1,904,324 semiprimes ≤ 1e7
• 17,427,258 semiprimes ≤ 1e8

Repository structure

paper/          — paper.tex, cover_letter.tex, paper.pdf (canonical)
paper/drafts/   — earlier draft versions
scripts/        — all Python and shell scripts
figures/        — all generated plots
datasets/       — raw data CSVs and JSON
results/        — CSV outputs from verification script