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 \(p\cdot R(p)=p(2p-1)=T(2p-1)=T(R(p))\), 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. (Independent verification: exhaustive two-pointer digit automaton to 400 digits; full search to \(10^8\); the palindromic channel is closed to \(10^{40}\) via Pell roots of square triangular numbers.)
Full write-up: [PDF / Zenodo DOI — link here after 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 of this sequence — after approval]
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 — after approval]
S3 — The general sequence: 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:
| Species | Count | Status |
|---|---|---|
| Trailing-zero terms (reduce to a weighted case) | 34 | reduction law |
| Family A — 39…97 / 79…93 | 14 | proved identity; all members, prime or not |
| Family B — 79…98 / 89…97 | 14 | proved: \((8\cdot10^m-2)(9\cdot10^m-3)=T(12\cdot10^m-4)\) |
| Palindromes | 2 | only 1 and 6 below 10⁸ (Pell roots of square triangulars) |
| Sporadic mirror pairs | 20 | open problem |
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 sporadic pairs below 10⁸ — (116, 611), (507, 705), (38517, 71583), (175868, 868571), (416024, 420614), (1181373, 3731811), (2041533, 3351402), (5099616, 6169905), (26129445, 54492162), (44067594, 49576044) — obey none of the known families. Whether they are infinite, and whether they hide further parametric families, is open.
3. 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.
4. 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
b-files, verification scripts and the digit automaton: [repository link]
5. 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.