Predictable Pattern of Non-primes

A hobby project, not a publication — an illustration rather than a result: how grouping the primes into intervals of the form

$$\frac{(x-1)^2}{2} < P < \frac{x^2}{2}$$

and drawing each interval as a centred horizontal segment produces a fan of vertical "lines of no primes". Every prime $P$ in row $x$ sits at

$$\frac{x^2}{2} - \frac{x}{2} - i = P$$

so since $\frac{x^2}{2} - \frac{x}{2} = T(x-1)$, a row is nothing but the triangular number $T(x-1)$ minus the column index $i$. A column can only be empty at a triangular index $i = k(k-1)/2$, where the row value factors as $P = (x-k)(x+k-1)/2$; for $x \ge k+3$ both factors exceed 1, so every such row is composite. The only primes a triangular column can hold sit in its two lowest rows: $x = k+1$ gives $P = k$ and $x = k+2$ gives $P = 2k+1$. A triangular column is therefore prime-free exactly when $k$ and $2k+1$ are both composite — so $i = 6, 45, 66, 120, \dots$ are empty, while $i = 3, 10, 15, 21, 28, \dots$ are not ($k$ or $2k+1$ prime there).

The mirrored half behaves differently. For $i &lt; 0$ the row value is $T(x-1) + |i|$, a sum, so nothing forces a factorisation and no column is prime-free. This asymmetry is structural, not visual: it follows from triangular numbers being non-negative.

The investigation started from the classical prime arrangements — the Klauber triangle and the Ulam spiral — and from arranging primes along an Archimedes spiral at varying angular rate.

Status and prior work

Read this before citing anything here. The 2023 text is unchanged, and it presents as new something that is largely classical. A literature check in October 2026 established:

• The identity $T(x-1) - T(k-1) = \frac{(x-k)(x+k-1)}{2}$ and the counting of such representations is classical: Glaisher 1907; Sylvester's enumeration (see OEIS A001227, A136107); T. Verhoeff, "Rectangular and Trapezoidal Arrangements", J. Integer Sequences 2 (1999), #99.1.6; M. A. Nyblom, "On the Representation of the Integers as a Difference of Nonconsecutive Triangular Numbers", Fibonacci Quarterly 39(3) (2001) 256–263.
• The composite statement itself is stated verbatim in OEIS A089306 (comment by Amarnath Murthy, 2003): "the difference of two triangular numbers is composite if the indices differ by more than 2."
• The general mechanism — a quadratic that factors over the integers yields a prime-free line, with a possible exception when a factor equals 1 — was already in Wikipedia, "Ulam spiral" (§Explanation) by 2022, and the square-analogue for the Klauber triangle in OEIS A005563 (comment by Charles Kusniec, 2017).
• The same conclusion, derived with the same $1+8T$ discriminant argument, was independently published in 2026: giammirove/triangular-prime-pyramid.

What is original here is the picture: this particular centred half-square array, in which the classical structure becomes visible as columns. The paper's abstract also promises a discussion of $\pi(x)$ that the text does not contain.

Two corrections a reader should apply to the 2023 text: the lemma in §2 is false as stated (it omits the case $x-k \le 2$), and the figure names in the .tex use the wrong case for Linux builds. ERRATA.md lists all six corrections for readers who only look at the compiled PDF. The correct statement is: for $i = k(k-1)/2$ and $x \ge k+3$, $P$ is composite, with primes possible only at $x = k+1$ ($P = k$) and $x = k+2$ ($P = 2k+1$).

Verified against a sieve for $x \le 20,000$ ($|i| \le 4,000$):

Fact	Result
Prime-free columns	35, all with $i \ge 0$, all triangular
Triangular columns in range	58 — the other 23 contain a prime in their bottom rows
Rule "prime-free $\iff$ $k$ and $2k+1$ both composite"	0 mismatches
Columns with $i < 0$ (mirrored half)	4 000 tested, all 4 000 contain a prime

So the mirrored half has no prime-free columns whatsoever. The 2026 publication giammirove/triangular-prime-pyramid states the composite-row result for its own pyramid, but that pyramid has no mirrored half — its rows run $r \ge 0$ only — so the one-sidedness never arises there. It is an addition relative to that text, and an elementary one: a difference of two triangular numbers needs non-negative indices, a sum does not.

Contents

File	Description
PrimePattern.tex	Paper source (LaTeX, article)
PrimePattern.pdf	Compiled paper
PatternAnalysis.m	Analysis of the observed pattern
PrimePattern.m	Generates the prime arrangements
PrimCountFunc.m	Prime counting / interval enumeration
FitS.m	Fit of the summation term
theorem.m	Numeric check of the theorem
*.png	Figures included by the paper
Psum.mat, xydata.mat	Data produced by the MATLAB scripts

Building

pdflatex PrimePattern.tex

Figures are referenced by relative path, so build from the repository root.

Provenance

Written between 22 September and 5 October 2023; published here unmodified apart from repository housekeeping. Per-file original modification timestamps are recorded in MANIFEST.md, and the 2023 content is committed with its original date. Written for curiosity, never submitted anywhere.

Author

Twan Mennink