The congruential obstructions to a D(n) pair of triangular numbers are exactly n ≡ 2,5 (mod 9)
Abstract
A D(n) pair of triangular numbers is a pair (Tₐ,T_b) with TₐT_b+n a perfect square, where Tₐ=a(a+1)/2. Bagchi and Zhou-Zheng prove that no such pair exists when n ≡ 2,5 (mod 9), observe computationally that n=12,17,42 also admit no pair with a ≤ 10⁵, conjecture that infinitely many such sporadic n exist, and assert that they "do not arise as a result of a congruential obstruction". The theorem they offer in support of that assertion is restricted to moduli m with 9 ∤ m — that is, it says nothing at exactly the moduli where their own obstruction lives. We remove the restriction at the prime 3: for every n ≢ 2,5 (mod 9) and every k, the congruence TₐT_b+n ≡ c²pmod3ᵏ is solvable with distinct positive indices, by one explicit Hensel lift together with a five-case split on n mod 27. Hence solvability modulo every power of 3 holds if and only if n ≢ 2,5 (mod 9), the congruential obstructions to a triangular D(n) pair are exactly those two classes, and the sporadic values 12,17,42 carry no obstruction at any modulus. Read together with a recent preprint of Bliznac Trebješanin, which proves that no D(12) pair exists at all, this exhibits n=12 as a failure of the local-global principle for the equation. We also record a per-index obstruction with certified two-modulus sieves, which cut the admissible first indices for n=12,17,42 to densities 91/675, 1/15 and 7/60, and we note that the printed list of n ≤ 50 without a pair omits n=50, which the source's own theorem excludes. All theorems below are machine-checked in Lean 4; S[sec:verif] says exactly what is checked and what is not, and S[sec:comp] collects the numerical observations that lie outside the formal development.
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Claim ledger
Stated results
TD1routine2026-09-03
tri, 2 Tₐ = a(a+1), and the periodicity Tₐ mod m depends only on a mod 2m
TD2known2026-09-03
The source's Theorem: no D(n) triangular pair for n = 2, 5 (mod 9) – proved in the stronger form that the congruence already fails modulo 9
TD3correction2026-09-03
n = 50 is missing from the source's printed list of n <= 50 with no pair, and its count 'thirty-six' should be thirty-five
TD4routine2026-09-03
Hensel at 3: an integer congruent to 1 mod 3 is a square modulo every power of 3, by an explicit lift
TD5candidate2026-09-03
For every n outside 2, 5 (mod 9) and every k, Tₐ T_b + n = c² is solvable mod 3ᵏ with distinct positive indices
TD6candidate2026-09-03
Solvability modulo every power of 3 holds if and only if n is not 2 or 5 mod 9 – with the source's 9-does-not-divide-m theorem, the congruential obstructions are exactly those two classes
TD7candidate2026-09-03
The sporadic n = 12, 17, 42 are solvable modulo every power of 3, hence have no congruential obstruction at any modulus
TD8routine2026-09-03
Controls for the local analysis at 3: n = 2 is unsolvable already mod 9, and the Hensel hypothesis is sharp
TD9routine2026-09-03
The per-index local obstruction: if Tₐ mod m admits no (b,c) mod m then Tₐ lies in no D(n) pair with b unbounded; its quadratic-residue form; and the source's 'a, b = 1 (mod 3)' remark
TD10routine2026-09-03
Certified sieves for n = 12, 17, 42: two verified bad-residue lists per n, and admissibility depending only on a mod 1350 / 1050 / 1800
TD11routine2026-09-03
Exactly 182 of 1350, 70 of 1050 and 210 of 1800 residue classes are admissible for n = 12, 17, 42 – densities 91/675, 1/15, 7/60
TD12routine2026-09-03
Controls for the sieve: a residue that is not bad, admissible indices exist, and the machinery does not reject n = 1
TD13routine2026-09-03
Controls for the source's theorem: its own Fermat example T1,T2,T15, a D(8) pair, a solution mod 3 for n = 2, and the three sporadic n lying outside the obstructed classes
TD14measurement2026-09-03
Sporadic census: 942 of the 7778 values n <= 10⁴ with n not 2, 5 mod 9 have no D(n) triangular pair with min index <= 10⁵; the sporadic rate rises monotonically with the number of small primes modulo which n is not a square, and is 0 for the 214 n that are squares mod every one of 3,5,7,11,13,17,19
This ledger entry is reported in prose and is not bound to a Lean theorem.TD15measurement2026-09-03
For n = 12, 17, 42 no D(n) triangular pair has an index at most 10⁷ – 100x the source's box, with the second index unbounded
This ledger entry is reported in prose and is not bound to a Lean theorem.TD16measurement2026-09-03
Compute-first gate: the source's n <= 50 run reproduces as 35 with a pair and 15 without, the control n = 1 gives 942 pairs in the scout's box, and the O(a^(1/2)) test agrees with direct search on 7200 cells
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. arXiv:2608.27697v1, Sounak Bagchi and Christian Zhou-Zheng, *Diophantine m-tuples of Triangular Numbers* (2026-08-27, v1 only; an Euler Circle project, also given as a JMM 2025 talk). Founding journal: journal/2026-09-03-triangular-dn-sporadic-founding.md.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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