Generalized Lucas Theorem

2025/02/12 by Hirsh, Jordan
#05A10 (Secondary) #11B65 (Primary) #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2502.19427

Abstract

Let p be a prime. Let A and B, A ≥ B ≥ 0, be integers with base p expansions A = α_iα_i-1… α₀ and B = β_iβ_i-1… β₀. Lucas proved that \binomAB ≡ ∏_j=0^j=i\binomα_jβ_j mod p. Similarly as proved by Kummer, the p-adic valuation v_p\binomAB is the number of borrows when computing A-B in base p, or the number of carries in (A-B)+B in base p. Davis and Webb discovered a generalization of Lucas's Theorem for prime powers. We prove a similar generalization in a different form using the concept of pseudo-digits.

Related

• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• Variations of Lucas' Theorem Modulo Prime Powers
• Partial Lucas-type congruences
• The p -adic Valuation of Lucas Sequences When p Is a Special Prime
• An extension of Lucas Theorem