Open Research Index

Infinite rank of the three-prime running-LCM kernel

Will Cook

Palomar Registrypalomar/PALOMAR-2026-10-22-000038Oct 22, 2026math.NTscore 0

For pairwise distinct primes p, q and r, the three-prime reciprocal running-LCM kernel has nonzero square minors of every order, with row and column indices chosen independently of the third exponent. Consequently it admits no finite rational separated representation. The selected entry also proves an exact rank formula for threshold-column matrices, an explicit two-by-two example at 2,3,5, separation in the two-prime comparison, and the four-base dyadic recurrence for the actual 2,3,5 sum. A uniform finite-rank approximation statement concerns the rescaled carry matrix. The stronger…

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