{"id":"palomar/PALOMAR-2026-10-18-000007","url":"https://github.com/AItoBit/erdos-684-elementary-bounds/tree/a59cf65171b19dc7785b767d3b88ad8ef754f3eb","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-18-000007-v1.json","repo":"https://github.com/AItoBit/erdos-684-elementary-bounds","tags":["math.NT","msc:11B65","msc:11N05","msc:11A51","number-theory","binomial-coefficients","erdos-problems"],"score":0,"title":"Elementary bounds for Erdos problem 684","trust":"high","domain":"math.NT","source":"palomar","authors":["AItoBit"],"reviews":0,"subject":"math.NT","summary":"Let U(n,k) be the full prime-power part of the binomial coefficient n choose k supported on primes at most k. For all sufficiently large n, there exists k at most n with k at most 514(log n)^2 and U(n,k) greater than n squared. For arbitrarily large n greater than one, U(n,k) equals one throughout k at most (log n)/8. This independent Mathlib-only implementation adapts the known residue averaging and prime-power obstruction of APSSV26 with coarse Chebyshev constants. It does not formalize that paper's sharper coefficients or Bae's stronger registered lower bound. The optimal order remains…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-18","reactions":{},"formalized":true,"formalizes":[{"title":"Short proofs in combinatorics and number theory","identifier":"https://arxiv.org/abs/2603.29961v2","relationship":"adapts"}]}