2014/06/06 by Edinah K. Gnang, Maksym Radziwiłł, Gnang, Edinah K. +3
Computer Science · Mathematics · #Advanced Mathematical Identities #Coding theory and cryptography #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1406.1704
openalex publication_date 2014/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to n as n goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to n and using exactly k multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers n, we compare the lengths of the arithmetic formulas for n that each encoding produces with the length of the shortest formula for n (which we estimate from below). We briefly discuss the time-space tradeoff offered by each.