Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1608.06834

Friends of 12

Doyon Kim
View original source ↗Read source document ↗Export metadata ↓

Abstract

A friend of 12 is a positive integer different from 12 with the same abundancy index. By enlarging the supply of methods of Ward [1], it is shown that (i) if n is an odd friend of 12, then n=m^2, where m has at least 5 distinct prime factors, including 3, and (ii) if n is an even friend of 12 other than 234, then n=2*(q^e)*(m^2), in which q is a prime greater than or equal to 29, e is a positive integer, and both q and e are congruent to 1 mod 4, and m has at least 3 distinct odd prime factors, one of which is 3, and the other, none equal to q, are greater than or equal to 29.

Explore related subjects

math.HOmath.NT

Keep this discovery

BibTeXRIS

Doyon Kim. 2016-08-19. Friends of 12. https://arxiv.org/abs/1608.06834

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

KEEP EXPLORING

Related papers

arXiv2026

Perspectives on the unit distance problem

This is a survey on an old open problem in combinatorics called the unit distance problem, and the field of mathematics around it, called incidence geometry. What do we know about the problem? Why is it difficult? How does it connect with other parts of math?

math.HO↗
arXiv2026

A Categorical Approach to Euclidean Ratios and Proportions

A categorial approach to the non-metric geometry in Books V and VI of Euclid's \textit{Elements} is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.

math.HO↗
arXiv2026

Some Early Results by Tutte Regarding the Cycle Double Cover Conjecture in 1948

OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture. Most media reports have characterized it as a 50-year-old open problem. In reality, according to a 1987 letter from Tutte to Fleischner, the Cycle Double Cover Problem has been open for at least 80 years. Two early results regarding the CDC conjecture were established in one of Tutte's 1949 publications.

math.HO↗