Bits of \(3^n\) in binary, Wieferich primes and a conjecture of Erdős
The authors of the present paper study the distribution of digits (bits) in the sequence of \(3^n, n=0, 1, 2, \ldots\), in binary expansion with generalizations, and in connection with the Erdős conjecture that: for \(k>8, 2^k\) is not the sum of distinct powers of 3 (published in [\textit{P. Erdős}, Math. Mag. 52, 67--70 (1979; Zbl 0407.10001)]). The authors start with the following conjecture: For all distinct primes \(p\) and \(q\) and every \(a \in {0, \ldots,q-1}\), let \(d_{n}(a)\) denote the number of \(a^{,}\)s appearing in the expansion of \((p^n)\) in \(q\)-ary number system \((p^{n})_{q}\). Then \[ \lim_{n \to \infty}\frac{d_{n}(a)}{\log_{q}p}=\frac{1}{q}. \] They prove results and the first proved result is that \[ \lim_{N \to \infty}\frac{1}{N}\sum^{N}_{n=1}\frac{d_{n,m}(a)}{m}=\frac{1}{h_{m}}\sum^{h_{m}}_{n=1} \frac{d_{n,m}(a)}{m}, \] where \(d_{n,m}(a)\) means the number of cases \(a \in {0, \ldots,q-1}\) and \(h_{m} = \sharp H_{m}\), \(H_{m}\) is the multiplicative group generated by \(p\) in \((\mathbb{Z}/q^{m})^{\times}\). They study the Wieferich primes (prime \(q\) is Wieferich prime, if \(2^{q-1}\equiv 1\pmod {q^2}\) in the following generalized form: The prime \(q\) is \(p\)-Wieferich at \(r\), \(r\) positive integer, if the multiplicative group generated by \(p\) modulo \(q^r\) is isomorphic to the group generated by \(p\) modulo \(q^{r+1}\). They prove that for different primes \(p\) and \(q\) the number \(n\), for which \(q\) is \(p\)-Wieferich at \(n\), is limited. The authors notice that the first \(m\) digits in the sequence \((p^n)_{q}=a_{0}+a_{1}q+\dots+a_{n}q^{n}\) is periodic in \(n\). Let \(K_{m}= \ker (H_{m} \rightarrow H_{m-1})\). The last proved theorem is: Let \(\bar{k}_{j}= \sharp K_{j}-1\). Then \[ \lim_{n \to \infty}\frac{1}{n(q-1)} \sum^{n}_{j=1}\bar{k}_{j} = 1. \]
scientific article; zbMATH DE number 6479850
scientific article