E124: Klauber triangle structure

size 101: 10,201 numbers and 1,252 primes

size 201: 40,401 numbers and 4,236 primes

size 301: 90,601 numbers and 8,769 primes

size 401: 160,801 numbers and 14,752 primes

size 999: 998,001 numbers and 78,359 primes

Prime counts shown are \(\pi(\text{size}^2)\), i.e., the number of primes \(\le \text{size}^2\).

Tags: number-theory, quantitative-exploration, visualization
See: Valid Tags.

Highlights

• A triangular “prime map” where row \(m\) contains the integers from \((m-1)^2+1\) to \(m^2\).

• Parameterizable run: vary --size from the command line (default: 301).

• Writes reproducible artifacts (params.json, report.md, and figures) into out/e124/.

Goal

Render the Klauber triangle (a square-number-based triangular arrangement) and highlight primes to reveal linear “streaks” and other geometric structures, then compare how that structure changes as size increases.

Background (quick refresher)

What is the Klauber triangle?

The Klauber triangle arranges the natural numbers into rows indexed by \(m=1,2,3,\dots\) such that row \(m\) contains the consecutive integers

\[(m-1)^2+1,\ (m-1)^2+2,\dots,\ m^2.\]

Each row has \(2m-1\) entries, and the triangle up to row size contains the integers \(1,\dots,\text{size}^2\). When primes are highlighted, surprisingly strong visual streaks can appear. See [Hill, 2016].

Optional background pages

• Prime numbers refresher

• Quadratic polynomials (algebraic) refresher

• Euler’s prime-generating polynomial refresher

Research question

Which prime-rich streaks are visible in the Klauber triangle for a given size, and how do those structures change as size increases?

Why this qualifies as a mathematical experiment

• Finite procedure: build a finite triangle window containing integers \(1,\dots,\text{size}^2\) and run a deterministic primality test.

• Observable(s): streaks/lines, local clustering, and how visibility changes with size.

• Parameter space: size (and, if you extend it, rendering/annotation choices).

• Outcome: figures and a short report capturing the key observation and caveats.

• Reproducibility: outputs saved to out/e124/ with a parameter snapshot.

Experiment design

• Computation: place integers \(1,\dots,\text{size}^2\) into the Klauber triangle row rule; mark primes.

• Coordinates: x is the integer column offset from the center column (0 at the center); the vertical axis is the row index \(n\) with \(n=1\) at the top.

• Prime classification: primes are computed using a sieve up to \(\text{size}^2\).

• Outputs: one or more figures showing prime positions in the triangle.

• Artifacts written:

  • figures/fig_01_klauber_triangle*.png (main figure) and figures/e124_hero_<size>.png (published hero)

  • params.json

  • report.md

Plot axes and coordinate conventions

The triangle is indexed by a row number and a horizontal offset within that row:

• y-axis: the row index \(m\in\{1,\dots,\text{size}\}\) (with \(m=1\) at the top)

• x-axis: the integer column offset from the center of the row:

  \[x\in\{-(m-1),\dots,(m-1)\}\]

For a given row \(m\) and offset \(x\), the corresponding integer in the Klauber triangle is:

\[ n(m,x)=(m-1)^2 + \bigl(x+(m-1)\bigr) + 1. \]

This formula matches the construction “odd squares on the left edge”: the leftmost entry in row \(m\) is \(n(m,-(m-1))=(m-1)^2+1\), and the row increases by 1 as \(x\) moves to the right.

If the implementation renders the triangle as a 2D array with Matplotlib imshow, the extent should be chosen so that tick labels correspond to the displayed \((x,m)\) values. When origin="upper" is used, \(m\) increases downward visually; in that case, interpret the y-axis tick labels accordingly (or flip the axis via the extent).

How to run

make run EXP=e124

Override the triangle size:

make run EXP=e124 ARGS="--size 501"

Direct invocation (always works):

uv run --extra dev python -m mathxlab.experiments.e124 --out out/e124 --size 501

Notes / pitfalls

• Larger size means more numbers (\(\text{size}^2\)) and a longer run time.

• “Looks-true” trap: visual regularity suggests hypotheses but does not prove theorems about primes.

• Rendering choices (marker size, interpolation) can hide or exaggerate streaks.

Extensions

• Quantify streaks: compute prime density along selected triangle-aligned lines and plot the densities.

• Highlight the square boundaries (\(m^2\)) or row indices to relate features to the row rule.

• Compare multiple sizes side-by-side (same styling and axis conventions).

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Reproduce:

make run EXP=e124

Parameters

• size: 301 (odd); visualizes integers 1..90601.

Notes

• The triangle arranges row n as (n-1)^2+1 .. n^2 (row length 2n-1).

• Prime-rich lines are linked to quadratic polynomials, similar to the Ulam spiral.

• This experiment is deterministic; seed does not change the output.

params.json (snapshot)

{
  "size": 301
}

References

• Klauber triangle definition and examples: [Hill, 2016].

• Exposition connecting multiple prime visualizations (includes Klauber and Sacks): [Brockmann, 2019].

• Recreational-math context and “pattern traps”: [Gardner, 1983], [Hoffman, 1989].

See also References.

Related experiments

• E024: Ulam spiral structure (Ulam spiral structure)

• E125: Sacks spiral structure (Sacks spiral structure)

• E126: Hexagonal number spiral structure (Hexagonal number spiral structure)

• E127: Quadratic prime-run atlas (n^2 + a n + b) (Quadratic prime-run atlas (\(n^2 + a n + b\)))

• E128: Quadratic modular obstructions (Euler-type) (Quadratic modular obstructions (Euler-type))

• E129: Euler lucky constants for n^2 + n + b (Euler lucky constants for \(n^2 + n + b\))

Parameters (example)

• size: 301 (triangle rows: 301, integers 1..90,601)

Recommended size range

The Klauber triangle up to row size contains \(\text{size}^2\) integers, so runtime and memory scale roughly with \(\text{size}^2\).

A practical range that works well for most runs:

• Minimum (still meaningful): size = 101

• Default / recommended: size = 301

• Comfortable upper range on typical laptops (varies): size = 999

• Above ~size = 1501: expect noticeably higher runtime (and potentially memory pressure), depending on the machine and the primality implementation.

Rule of thumb: start with 301, then try 501, 701, 901. Increase further only if runtime remains acceptable.

Summary

We render the Klauber triangle containing the first 90,601 positive integers and highlight primes. Even at this moderate scale, visually prominent streaks appear: some triangle-aligned lines contain noticeably more primes than nearby lines.

Key observations

• Several streaks stand out strongly against a background that still looks relatively “noise-like”.

• The most visible structures shift as size increases; some streaks weaken while others become clearer.

Interpretation

The Klauber triangle is a structured re-indexing of the integers where many visually straight features correspond to simple polynomial or arithmetic progressions in the underlying index. This makes it a useful “pattern detector” for prime-rich sequences, but it is not evidence of a global rule.

Caveats

• Finite window: patterns can look stronger/weaker depending on the chosen size and zoom level.

• Visualization choices matter: marker size and interpolation can hide or exaggerate streaks.

• Different triangle conventions (if you change the row rule) change which streaks appear.