Quartic Divisibility by Thirty-Six
Abstract
The quartic 27k^4+108k^3+171k^2+126k+36 is divisible by 36 for every integer k.
Theorem 1.1 (Thirty-six divides the quartic for every integer).
Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/QuarticThirtySix.thirtySix_dvd_m (✓ std3). ∎
Source. Repository-derived.
Commentary.
The quartic m(k) = 27k^4 + 108k^3 + 171k^2 + 126k + 36 is divisible by 36 for every integer k. Reducing modulo 36, the polynomial evaluates to zero on every residue class, so 36 divides m(k) identically. The residue check is a finite kernel decision over the 36 elements of ZMod 36, lifted to the integers by the standard cast-vanishes-iff-divides equivalence.
This is the self-contained arithmetic corroboration of the 36-theorem; it makes no claim about the geodesic-word or fixed-point-form context in which the quartic arises.
References
- Truth anchor:
D5/S3/Arith/Congruence/QuarticThirtySix.thirtySix_dvd_m