2021/05/31 by Samer Seraj, Seraj, Samer
Computer Science · #11A07 (Secondary) #11A51 #11A63 (Primary) #Chaos-based Image/Signal Encryption #Cryptography and Data Security #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2106.00382
openalex publication_date 2021/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Notice that the square of 9376 is 87909376 which has as its rightmost four digits 9376. To generalize this remarkable fact, we show that, for each integer n≥ 2, there exists at least one and at most two positive integers x with exactly n-digits in base-10 (meaning the leftmost or nth digit from the right is non-zero) such that squaring the integer results in an integer whose rightmost n digits form the integer x. We then generalize the argument to prove that, in an arbitrary number base B≥ 2 with exactly m distinct prime factors, an upper bound is 2m -2 and a lower bound is 2m-1-1 for the number of such n-digit positive integers. For n=1, there are exactly 2m -1 solutions, including 1 and excluding 0.