The Triangular Mirror-Prime Theorem, its sequences, and its constant

1. The theorem

Write \(R(p)\) for the decimal digit reversal of \(p\), and \(T(k)=k(k+1)/2\) for the \(k\)-th triangular number.

The members are 37, 73, 397, 7993, 799993, 7999993, … The forward direction is a one-line identity: if \(R(p)=2p-1\) then

and the two digit families are exactly the mirror pair satisfying that relation.

Proof outline (converse)

• From \(2p\,R(p)=k(k+1)\) and \(p\) prime, \(p\mid k\) or \(p\mid k+1\); the size bound \(R(p)<10p\) forces the cofactor to be at most 4.
• Cofactor 3 dies modulo 9 (digit reversal preserves digit sums); cofactor 4 dies by a last-digit parity argument.
• The surviving relations reduce to two digit equations. \(R(x)=2x-1\) is solved completely by D. G. Radcliffe, Numbers that are nearly doubled when reversed (2015): one solution per length, the pattern 39…97.
• \(R(x)=2x+1\) has no solutions in any length: the extreme digit columns force \(x = 3\,\square\,6\), and the equation descends from length \(d\) to length \(d-2\), reaching a contradiction.

The last bullet used to be supported by an exhaustive search to 400 digits. It no longer rests on a search depth: \(R(x)=2x+1\) is the pencil \(1\cdot R(k)=2k+1\), and the decision procedure of §3 returns empty with a certificate — its bounded state graph has a single state and an empty live subgraph, so there is no solution at any length. Run against \(R(x)=2x-1\) the same machine returns infinite and reproduces Radcliffe's family from scratch: 1, 37, 397, 3997, 39997, …

Full write-up: [PDF / Zenodo DOI — pending deposit] · [arXiv — after submission]

2. The sequences

S1 — Primes p such that p·R(p) is triangular

37, 73, 397, 7993, 799993, 7999993, 79999999993, 7999999999993, 79999999999993, 399999999999997, 399999999999999999997, …

Exactly the primes in the two theorem families; equivalently, the primes of OEIS A083818 together with the primes of A169830. Exponent sequences: OEIS A101398 (4·10ⁿ−3 side) and A099190 (8·10ⁿ−7 side). All 25 members up to 873 digits certified prime (APR-CL). [OEIS A-number — submission pending]

S2 — Primes p such that p + R(p) is triangular

3, 5, 23, 41, 2273, 2543, 2633, 2903, 4091, 4271, 4451, 4721, 61979, 62969, 68909, 84947, 86927, 87917, 206933, …

145 terms below 10⁷, reaching only 12 triangular targets. Structural result (“signature law”): all terms hitting the same target T(m) share the same vector of digit-pair sums \(s_i = x_i + x_{d-1-i}\). For even digit counts, \(p+R(p)\equiv 0 \pmod{11}\). [OEIS A-number — submission pending]

S3 — OEIS A397674: n·R(n) triangular, any integer n

1, 6, 10, 37, 73, 78, 87, 116, 397, 507, 611, 705, 793, 798, 897, 1200, 2100, 3230, 3997, 7993, 7998, 8997, …

The primes of S1 are exactly the prime subsequence. Census of the 84 terms below 10⁸:

The sequence is closed under digit reversal (the product does not care which mirror you hold), so every non-palindromic, non-trailing-zero term comes in a mirror pair. The twenty sporadic terms below 10⁸ form ten such pairs and obey none of the known families. Until August 2026 their status was the standing open problem of this page; §3 settles it.

Families I and II were called “Family A” and “Family B” in the July 2026 version of this page.

3. The decision procedure

Every term of A397674 that is neither a palindrome nor a multiple of ten satisfies exactly one pencil — a linear digit-reversal equation

A · R(k) = B · k + γ

attached to a reduced fraction \(u/v\) and a sign \(\sigma\in\{-1,+1\}\) by

g = 2 if u is even else 1
A = u²/g        B = 2v²/g        γ = σ·uv/g

This reframes the problem. “Sporadic” was never a property of a term; it is a property of its pencil. A term looks sporadic exactly when its \(u/v\) sits high in the Farey order — far enough out that a sweep of the small grid never reaches it. The pencil of 26129445, for instance, is \(u/v = 2885/2946\).

The automaton, and why empty is a theorem

A pencil is decided by an automaton that reads the digits of \(k\) from both ends inward. One step consumes the leftmost and rightmost remaining digits, resolving two columns of the equation at once, because a single digit pair contributes to both a low position and a high position. The state is a pair \((c,e)\): the carry travelling inward from the units end, and the value accumulated inward from the leading end. The ends meet in the middle, where acceptance reads \(c+e=0\) for even lengths and \(c+(A-B)w+10e=0\) for odd lengths with middle digit \(w\).

The state space is made finite by a confinement box \(X=\max(A,-\gamma)\), \(Y=\max(B,\gamma)\), with \(-Y\le c\le X\) and \(-X\le e\le Y\). That box is the load-bearing wall of the whole construction, and it is proved sound in two parts, because the two variables behave differently:

• Lemma C. The interval \([-Y,X]\) is forward invariant for the carry: dividing by ten at each step contracts faster than a digit column can push. The bound never even fires.
• Lemmas E1–E2. The accumulator \(e\) can leave \([-X,Y]\), since its recursion multiplies by ten. But escape is absorbing — once \(|e|\) is outside it grows forever — and acceptance requires being inside. So discarding escaped states loses no solution.

Therefore a verdict of empty means the equation has no solution at any digit length. It is not a report that a search went some number of digits deep. That distinction is the point: the pencil \(225\,R(k)=512k-240\) looks empty if you enumerate to 34 digits, and its smallest solution has 51.

On the live subgraph — states both reachable from a start and co-reachable to an accepting state — Tarjan's algorithm settles the rest. Empty live subgraph gives empty; acyclic gives finite, and enumerating the DAG returns the complete solution list; a directed cycle gives infinite, with the cycle itself as the pumping witness.

Scope: the automaton assumes \(k\) has no trailing zeros, so that \(R(k)\) has the same digit length. Trailing-zero terms reduce to shorter weighted instances and are handled separately.

The ten sporadic pairs, decided

Three infinite, seven finite, none undecided. The seven finite pencils are complete: the lists below contain every solution at every length.

Five of the ten pairs are provably solitary — no companion at any digit length, in either direction. The infinite pencils pump: writing \(\mu\) for the digit length of the smallest solution and \(p\) for the cycle length,

32·R(k) = 169k − 52     μ = 3,  p = 36
  116
  117323652188164483082676347811835516916                                     (39 digits)
  117323652188164483082676347811835516917323652188164483082676347811835516916 (75 digits)

8·R(k) = 9k + 6          μ = 8,  p = 5
  44067594
  4406759324067594          (16 digits)
  441789206758210794        (18 digits)
  440675932406759324067594  (24 digits)

2738·R(k) = 8649k + 3441 μ = 7,  p = 231
  1181373    (live subgraph: 67 737 states, 1 978 887 edges)

Every witness above was checked three ways: it satisfies its pencil in exact integer arithmetic; \(8kR(k)+1\) is a perfect square, so \(k\,R(k)\) really is triangular; and the nine-digit terms 565066386 and 683660565 appear in an independently generated census to 10⁹ with triangular index 878992155, matching the value computed here. The 39- and 75-digit witnesses lie far beyond the published b-file, which stops at 96055860.

The Farey sweep

Classifying every primitive pencil with \(u,v\le 40\) — 1958 of them — takes about 40 seconds:

N = 40   pencils = 1958
  EMPTY     1857
  FINITE    18
  INFINITE  83
outside slope window: 654, non-EMPTY among them: 0

The mirror involution \((u,v,\sigma)\mapsto(2v/u,-\sigma)\) is respected throughout: mirror pencils always agree, as they must, since \(k\) and \(R(k)\) stand or fall together.

What is still open

The question moved up a level rather than closing. It is no longer “do the sporadic terms hide further families?” — each is now a decided pencil. It is:

• Do pencil heights grow without bound among solvable pencils, or is there a largest height that admits a solution?
• How many solvable pencils have height \(\le H\)? An asymptotic here would say something real about the density of A397674.
• Is the slope corridor \([1/\sqrt5,\ \sqrt{20}]\) a theorem? Empirically all 654 pencils outside it at \(N=40\) are empty. This is recorded as a conjecture and is not used for pruning.
• The infinitude of the two theorem families — equivalently, of primes of the form \(4\cdot10^n-3\) and \(8\cdot10^n-7\) — remains untouched.

4. The constant

The sum of the reciprocals of the theorem’s primes converges extremely fast (members are doubly-exponentially sparse):

G = 0.043371033346144668501877952658711565159490576263…

All members through exponent 872 are certified primes (APR-CL), and Kamada’s exhaustive searches cover both families to exponents beyond 10⁵, so the digits of G are determined at least to order 10⁻¹⁶³⁸ (the next PRP member). The infinitude of the member set — equivalently, of primes in either family — is open.

5. Data and code

Everything on this page is reproducible from one predicate.

(PARI) is(p) = isprime(p) && ispolygonal(p*fromdigits(Vecrev(digits(p))), 3)

# Python
from sympy import isprime
from math import isqrt
def is_member(p):
    q = int(str(p)[::-1]); s = 8*p*q + 1
    return isprime(p) and isqrt(s)**2 == s

FareyMirror — the decision procedure, as a tool

The machinery of §3 is released as a small program. It decides any digit-reversal equation \(A\,R(k)=B\,k+\gamma\), not just the ones on this page, and it ships with the confinement proof that makes its empty verdict meaningful. Pure Python, standard library only, 81 tests.

$ tar xzf fareymirror-0.1.0.tar.gz && cd fareymirror-0.1.0
$ python3 -m fareymirror classify --A 18 --B 25 --gamma 15
18*R(k) = 25*k + 15
  box            X=18  Y=25   (-Y<=c<=X, -X<=e<=Y)
  live subgraph  9 states, 22 edges
  verdict        FINITE
  solutions      complete list:
                 507   (3 digits)
                 554262627402908967   (18 digits)

$ python3 -m fareymirror verify 116
pencil   = 32*R(k) = 169*k - 52   (u=8, v=13, sigma=-1)
verdict  = INFINITE

$ python3 -m fareymirror sweep 40 --outside

Contents: the decision core, the confinement proof (proofs/confinement.md), the decided pencils (docs/RESULTS.md), the conjectures kept deliberately separate from the theorems (docs/conjectures.md), and the test suite including the box-inflation reinforcement check. Code MIT, text CC BY 4.0. [public repository — mirror pending]

b-files and the earlier verification scripts are archived with the research package. [Zenodo DOI — pending deposit]

6. Provenance

The theorem grew out of an observation in Hebrew gematria: the letter values of Genesis 1:1 total \(2701 = 37\times 73 = T(73)\), the product of two mirror primes. Asking which other primes share that behaviour, and refusing every answer that could not survive adversarial testing — Monte Carlo controls, ablation, out-of-sample replication, literature search — led to the characterization above. The full story, including everything that was falsified along the way, will appear as an essay (Palimpsesto).

Research conducted with extensive computational assistance from Claude (Anthropic): search, verification, adversarial review and formalization support. All results were independently re-verified and are the author’s responsibility.

7. How to cite

8. Revision history

2026-08-11 — Added §3, the decision procedure, and released FareyMirror 0.1.0 with its confinement proof. All ten sporadic mirror pairs of A397674 decided: three infinite, seven finite, five provably solitary; complete solution lists for every finite pencil; witnesses of 39 and 75 digits. The \(R(x)=2x+1\) step of the converse no longer rests on a 400-digit search but on an empty certificate. S3 identified as OEIS A397674; families A and B renamed I and II; canonical URL fixed to math.k14.uk.

2026-07-03 — First published: theorem, proof outline, sequences S1–S3, census to 10⁸, and the constant G.