Erdős #269: exact three-prime running-LCM identity and infinite kernel rank
For pairwise distinct primes p, q, r and every nonzero x, the least common multiple of all smooth numbers p^i q^j r^k at most x equals the product of the three maximal pure prime powers below x. The reciprocal-height kernel K(i,j,k) built from that identity is constant on every logarithmic cell, and the first m positive jump values across the three prime channels number exactly 3m. Every finite exponent box sums to the height-fibre normal form, the sum over genuine running-LCM heights of the fibre cardinality divided by the height, and a multiplicative shell of width factor r inside a sorted…
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