The p-adic valuation of n! equals Σ_{k ≥ 1} ⌊n/p^k⌋. Equivalent form: v_p(n!) = (n − s_p(n))/(p − 1), where s_p(n) is the sum of base-p digits of n. Immediate consequence: v_p(C(m,n)) = (number of carries when adding n to m − n in base…
The p-adic valuation of n! equals Σ_{k ≥ 1} ⌊n/p^k⌋. Equivalent form: v_p(n!) = (n − s_p(n))/(p − 1), where s_p(n) is the sum of base-p digits of n. Immediate consequence: v_p(C(m,n)) = (number of carries when adding n to m − n in base…