{"id":"palomar/PALOMAR-2026-10-21-000019","url":"https://github.com/wcook04/plectis-erdos-lean/tree/c39cea6a976747d132a848d26f110b9c608d5a45","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-21-000019-v1.json","repo":"https://github.com/wcook04/plectis-erdos-lean","tags":["math.NT","msc:11B83","msc:11A05","number-theory","lcm","erdos-problems","multiplicative-number-theory"],"score":0,"title":"Erdős #269: exact three-prime running-LCM identity and infinite kernel rank","trust":"high","domain":"math.NT","source":"palomar","authors":["Will Cook"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-21","reactions":{},"formalized":true}