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2:37 PM
2
Q: Number of $1$'s in the ternary expansion of $a_p=\frac{3^{p-1}-1}{p}$

tanjiaI am considering a problem concerning Cantor set. Let $p>3$ be a prime. Is there a way to caculate the number of $1$ in the ternary expansion of $a_p=\frac{3^{p-1}-1}{p}$ or just to decide whether $1$ appears in the ternary expansion? Any result related to this quesion is also welcome.

0
Q: Residues distribution modulo an interval

ReverseFlowControlGiven a number $n$ and an Interval $I = [ \; \lfloor n^{1/4} \rfloor, \lfloor n^{(1/3) \rfloor \;} ]$, can we say anything about the distribution of $\{ n \mod b \;\;| \; b \in I \}$? In particular, if I wanted the residue to be close to any region in $[0, b-1]$, say close to the "top" of the re...

6
Q: Pair matching between divisors less and more than $\sqrt{N}$

thematdevLet $n$ be the positive integer. Let $A$ and $B$ be sets of divisors of $n$ less and more than $\sqrt{n}$ respectively. Consider bipartite graph $(A, B)$, where two vertices are connected when one divides another. Denote $M(n)$ number of perfect matchings in this graph. Is $M(n) > 0$ for all $n$(...

 

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