On a Conjecture of Cusick Concerning the Sum of Digits of n and n + t
(1509.08623)Abstract
For a nonnegative integer $t$, let $ct$ be the asymptotic density of natural numbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $ct > 1/2$ for $t$ in a set of asymptotic density $1$, thus giving a partial solution to a conjecture of T. W. Cusick stating that $c_t > 1/2$ for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial $X(X + 1)\cdots(X + t - 1)$ has less than $2t$ zeros modulo $2{t+1}$. The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function, using methods from analytic combinatorics.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.