PRIME FACTORIZATION AND NORMAL NUMBERS
Journal Title: Дослідження в математиці і механіці - Year 2015, Vol 20, Issue 2
Abstract
Research of the first author was supported in part by a grant from NSERC. Let 𝑞 > 2 be a fixed integer. Given an integer 𝑛 > 2 and writing its prime factorization as 𝑛 = 𝑝1𝑝2 ・ ・ ・ 𝑝𝑟 , where 𝑝1 6 𝑝2 6 ・ ・ ・ 6 𝑝𝑟 stand for all the prime factors of 𝑛, we let ℓ(𝑛) = 𝑝1 𝑝2 ・ ・ ・ 𝑝𝑟, that is the concatenation of the respective base 𝑞 digits of each prime factor 𝑝𝑖, and set ℓ(1) = 1. We prove that the real number 0.ℓ(1)ℓ(2)ℓ(3)ℓ(4) . . . is a normal in base 𝑞. In fact, we show more, namely that the same conclusion holds if we replace each 𝑝𝑖 by 𝑆(𝑝𝑖), where 0𝑆(𝑥) ∈ Z[𝑥] is an arbitrary polynomial of positive degree such that 𝑆(𝑛) > 0 for all integers 𝑛 > 1. We prove analogous results and in particular that, given any fixed positive integer 𝑎, the real number 0.ℓ(2 + 𝑎)ℓ(3 + 𝑎)ℓ(5 + 𝑎) . . . ℓ(𝑝 + 𝑎) . . . , where 𝑝 runs through all primes, is a normal number in base 𝑞. MSC: 11K16.
Authors and Affiliations
J. -M. De Koninck, I. Katai
Keywords
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