Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2201.00710

Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences

Rüdiger Jehn
View original source ↗Read source document ↗Export metadata ↓

Abstract

For numbers $x$ coprime to $10$ there exist infinitely many binary numbers $b$ such that the greatest common divisor of $b$ and rev($b$) = $x$ and the sum of digits of $b = x$ (rev($b$) is the digit reversal of $b$). In most cases, the smallest $b$ that fulfill these two constraints contain just a few zeros. But in some cases like for $x = 7, 11, 13$ and $37$, $b$ must contain more zeros than ones and these $b$ are called long solutions. For $11$ and $37$ it follows directly from the fact that these are porous numbers. For $7$ and $13$, the proofs that they have long solutions are presented in this paper.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Rüdiger Jehn. 2022-01-03. Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences. https://arxiv.org/abs/2201.00710

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗