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On the Binary Digits of a Power

✍ Scribed by Bernt Lindström

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
399 KB
Volume
65
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let B(m) denote the number of ones in the binary expansion of an integer m 2. We prove that lim sup m Ä B(m h )Âlog 2 m=h for integers h 2. We also prove the same result with m h replaced by any polynomial a 0 m h +a 1 m h&1 + } } } +a h with integer coefficients and a 0 >0.

1997 Academic Press

We may mention that the problem of bounding B(m h )ÂB(m) has been studied by Stolarsky [1], but this is a different problem.

2. PROOF

We shall use numbers m with noninterfering terms. Consider the simple examples a2 p \b with a, b 1 and b<2 p . Then terms are nontinterfering article no. NT972129

📜 SIMILAR VOLUMES

Algebraic Independence of the Power Seri
Algebraic Independence of the Power Series Defined by Blocks of Digits
✍ Yoshihisa Uchida 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 128 KB

Let q 2 be an integer and let w be a block of 0, ..., q&1 of finite length. For a nonnegative integer n, let e(w; n) denote the number of occurrences of w in the q-adic expansion of n. Define f (w; z)= n 0 e(w; n) z n . We give necessary and sufficient conditions for the algebraic independence of fu