An Extension of the Fundamental Theorem on Right-angled Triangles
نویسندگان
چکیده
{3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how many PPTs a given integer can be a member of. The answer to this simple question is, surprisingly, that a given integer n can play the role of a specified side in either 0 or 2k−1 different PPTs, where k is the number of distinct prime factors of n. Our result is a generalisation of what Fermat grandly called the Fundamental Theorem on right-angled triangles ([2], chapter 5), which states that:
منابع مشابه
A novel chiral phase of achiral hard triangles and an entropy-driven demixing of enantiomers.
We investigate the phase behavior of a system of hard equilateral and right-angled triangles in two dimensions using Monte Carlo simulations. Hard equilateral triangles undergo a continuous isotropic-triatic liquid crystal phase transition at packing fraction ϕ = 0.7. Similarly, hard right-angled isosceles triangles exhibit a first-order phase transition from an isotropic fluid phase to a rhomb...
متن کاملThe geometry of the curve graph of a right-angled Artin group
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen–Thurston classification for elements in the rig...
متن کاملEmbedability between right-angled Artin groups
In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph Γ, we produce a new graph through a purely combinatorial procedure, and call it the extension graph Γ of Γ. We produce a second graph Γ k , the clique graph of Γ, by adding extra vertices for each complete subgraph of Γ. We prove that each finite induced subgraph Λ of Γ gives ris...
متن کاملClassroom note: Almost-isosceles right-angled triangles
We provide an elementary method to show that there exist infinitely many right-angled triangles with integral sides in which the lengths of the two non-hypotenuse sides differ by 1. The method also enables us to construct all such right-angled triangles recursively. 1. Introduction There does not exist any isoceles right-angled triangle with integral sides. Does there exist a right-angled trian...
متن کاملOn lifting of biadjoints and lax algebras
Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{Ps}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $ then this lifting is also right biadjoint provided that $mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, ...
متن کامل