The prime pattern nobody could explain
In 2018 two amateurs found a striking image hidden in the prime numbers, published it, and said plainly that one half of it was beyond them. Nobody formalised it in the eight years since. Here is the proof of what it is — including the part they couldn’t explain — and an honest account of how much of it is really about primes at all.
What the picture is
Write the integers into a triangle: row 1 gets one cell, row 2 gets two, row R gets R. Each cell swallows n consecutive integers, where n is a width you choose. Colour a cell light if it contains at least one prime, red if it contains none.
For most widths almost every cell is light — primes are common enough that a block of a hundred consecutive integers nearly always contains one. The red cells are the interesting ones: whole blocks of integers with no prime anywhere inside. They are not scattered at random. They fall into sharp rays and lattices, and the discoverers noticed that the resulting figure looked like a known object from number theory. For even widths, they were right.
Why the pattern exists
The whole thing turns on one observation. A cell is not an arbitrary clump of integers — the way the triangle is laid out, the numbers inside a single cell in row R are spaced exactly R apart. They form an arithmetic progression with common difference R, which means every number in the cell leaves the same remainder when divided by R.
That settles almost everything. If the cell’s column number shares a factor with its row number, then every single integer in that cell is divisible by that shared factor — so none of them can be prime (barring trivially small cases), and the cell is red, guaranteed, forever. And when the column and row share no factor, the progression is exactly the kind that Dirichlet’s theorem populates with primes; make the width large enough and a prime shows up.
Theorem (even widths). As the width grows through even values, the pattern converges cell by cell to the table that marks (R, C) exactly when R and C share no common factor — OEIS A054521, the coprimality table.
This is what the discoverers saw and identified correctly. Press show predicted table above with an even width selected: the picture stops being about primes and becomes a statement about divisibility, and the two agree.
The half-twist: the part they couldn’t explain
At odd widths a different figure appears. It is clearly related to the first — same general character, same rays — but shifted, and the discoverers said they had no explanation for it. Flip the width above between 100 and 101 and you can watch it happen.
Here is the answer. Look again at the remainder each cell occupies. When the row number is even and the width is odd, the arithmetic of the layout shifts that remainder by exactly half the row — by R/2. Every other combination of parities leaves it alone. So:
Theorem (odd widths). The odd-width pattern is the same coprimality table, with every row R ≡ 2 (mod 4) rotated by half its length. Rows with R odd, or with R divisible by 4, are completely unchanged.
That is the entire mystery: a half-turn applied to one row in four, forced by a parity clash between the width and the row number. The rotated rows are exactly those where R/2 is odd, because that is precisely when adding R/2 changes which factors the column shares with the row.
Now the deflationary part
It would be easy to stop there and call this a hidden signature of the primes. It is not, and the controls are the honest way to show it.
Replace the primes with numbers that merely avoid the smallest few prime factors — anything not divisible by 2, 3, 5, 7, 11 or 13 — and the picture barely changes. On a width-74 triangle, 98.9% of cells are identical. All the rays, all the structure, essentially the entire image survives an operation that throws the actual primes away completely.
Replace them with a random set of the same density and the picture is annihilated. Every structured red cell fills in.
Try both above. The conclusion is unavoidable and worth stating plainly: the famous imagery is sieve structure, not prime mystery. It reflects which small numbers divide which others, which is why a set defined purely by small divisibility reproduces it. What is genuinely specific to the primes is a much thinner residue — a list of 21 exceptional cells, all in prime-numbered rows, the last of them at width 74, where a cell that should contain a prime happens not to.
You can catch that last exception in the act. Select width 74 with a prime marked: the figure has 1080 unmarked cells. Now press show predicted table: 1079. The single cell of difference is row 73, column 43 — a cell whose row and column share no factor, so the theory says a prime should turn up somewhere in its 74 integers, and none does. It is the last such accident anyone has found: we checked every width up to 420 and spot-checked to 600 without seeing another.
Where it stops being provable
The convergence theorem is pointwise: fix a cell, grow the width, and it eventually settles. Proving that the entire visible triangle is correct at any given width is much harder, because the deep rows involve primes in progressions with very large moduli inside short windows.
What can be proved unconditionally: every row of the triangle is at least one-sixth correct, so the exceptions can never exceed about a quarter of the figure. Assuming the Generalised Riemann Hypothesis, everything is exactly right out to rows of size about the square root of the width. Getting the full statement needs a bound on how primes distribute across many short windows at once that is, as far as we could establish, beyond current technology — we were able to isolate the exact missing ingredient, which is not the same as supplying it.
Empirically the picture is cleaner than any of that: past width 74 we found no exceptions at all, checking every width up to 420 and spot-checking to 600. Curiously the primes here are about nine times better behaved than the standard random model predicts, and we do not know why.
How this was checked
| Check | Result |
|---|---|
| Transform reconstructed from the discoverers' original code | reproduces their published figures, including a known quirk at width 14 |
| Every theorem re-derived from scratch by an independent adversarial session | confirmed; two minor errors found and fixed |
| All numerical claims recomputed with fresh code | exception list matched exactly; control figures reproduced to the cell |
| Prior art searched (the pattern relates to reduced residue systems mod primorials) | no prior formalisation found; A054521 has no prime connection recorded |
The two errors the referee caught are worth naming, since a post like this is only worth as much as its error-reporting: an illustrative example inside one proof was simply false (the proof itself stood), and a true inequality was justified by the wrong factorisation. Both are fixed in the paper.
The paper
A nine-page note with the full statements and proofs — the construction, the four theorems, the odd-width resolution, the control experiments and the two open conjectures — is here: The “Parallax Compression” prime pattern: exact structure, limit theorems, and the odd-width rule (PDF). All code is reproducible; the figures above are generated live in your browser from the definition, not loaded as images.
Credit where it is due
The pattern was found and published in 2018 by Nova Spivack and Shaun Gilchrist, who called it Parallax Compression, described it carefully, and were explicit about what they could and could not explain. Being clear about the boundary of your own understanding is what made the open question findable eight years later. The formalisation, the proofs and the controls here were produced by AI agent swarms (Claude, Anthropic) directed by Scry Visuals, with every result independently re-derived by a separate adversarial session before being recorded.
This post mirrors the page at scry.art/parallax, which also hosts the paper.