Christian Aebi

activity

20122026

most citedLattice equable quadrilaterals II -- kites, trapezoids and cyclic quadrilaterals

1 citations · 1 across the 15 of their papers we have counts for

collaborators

18 papers

math.GM2026

Lattice triangles whose centers are lattice points

Christian Aebi, Grant Cairns

We show that for an integer $ℓ$ , there exists an acute integer lattice triangle of lattice perimeter $ℓ$ such that its orthocenter is an integer lattice point, if and only if…

math.CO2026

The sad life of lattice triangles

Christian Aebi, Grant Cairns

This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circ…

math.CO2024

Equable Parallelograms on the Eisenstein Lattice

Christian Aebi, Grant Cairns

This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these p…

math.CO2023

Less than Equable Triangles on the Eisenstein lattice

Christian Aebi, Grant Cairns

We classify perimeter dominant triangles whose side lengths are in $3 N$ and whose area is in $\frac{3}{4} N$ . There is one exceptional example, which is equi…

math.CO2023

Following in Yiu's Footsteps but on the Eisenstein Lattice

Christian Aebi, Grant Cairns

Paul Yiu proved that all Heron triangles are realizable on the integer lattice. We give an analogous result for triangles with vertices on the Eisenstein lattice.

math.GM2023

Equable Triangles on the Eisenstein Lattice

Christian Aebi, Grant Cairns

We show that there are only two equable triangles having vertices on the Eisenstein lattice, up to Euclidean motions.

math.GM2022

A simple sum for simplices

Christian Aebi, Grant Cairns

We give a vector identity for $n + 2$ points in $R^{n}$ . It follows as a corollary that when $n$ is odd the sum of the signed volumes of the $n$ -simplices is zero, and when $n$ …

math.MG2021

Lattice Equable Quadrilaterals III: tangential and extangential cases

Christian Aebi, Grant Cairns

A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This…

math.MG2021

A vector identity for quadrilaterals

Christian Aebi, Grant Cairns

A surprising simple result about quadrilaterals is given as an application of the vector triple product identity.

math.GM2021★ 1 cited

Lattice equable quadrilaterals II -- kites, trapezoids and cyclic quadrilaterals

Christian Aebi, Grant Cairns

We show that there are 4 infinite families of lattice equable kites, given by corresponding Pell or Pell-like equations, but up to Euclidean motions, there are exactly 5 lattice eq…

math.NT2020

Lattice Equable Quadrilaterals I -- Parallelograms

Christian Aebi, Grant Cairns

This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the g.c.d. of the sid…

math.NT2017

The Quartic Residues Latin Square

Christian Aebi, Grant Cairns

We establish an elementary, but rather striking pattern concerning the quartic residues of primes $p$ that are congruent to 5 modulo 8. Let $g$ be a generator of the multiplicative…

math.NT2015

Sums of Quadratic residues and nonresidues

Christian Aebi, Grant Cairns

It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous res…

math.NT2015

Wolstenholme again

Christian Aebi, Grant Cairns

We give an elementary and self-contained proof of the equivalence of a collection of Wolstenholme-type congruences due to Helou and Terjanian.

math.NT2013

Wolstenholme and Morley, Primes and Pseudoprimes

Christian Aebi

In this note, we prove $p^{2}$ is a Morley pseudoprime (of order 2) iff $p^{2}$ is a Wolstenholme pseudoprime (of order 2) iff $p$ is a Wolstenholme prime iff $p$ is a Morley prime. Co…

math.NT2013

Morley's other miracle: $\displaystyle 4^{p-1}\equiv\pm (\smallmatrix p-1 \frac{p-1}{2} \endsmallmatrix) \pmod {p^3} $

Christian Aebi, Grant Cairns

Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle bi…

math.NT2013

Wilson Theorems for Double-, Hyper-, Sub- and Super-factorials

Christian Aebi, Grant Cairns

We present generalisations of Wilson's theorem for double factorials, hyperfactorials, subfactorials and superfactorials.

math.HO2012

Sylvester's, Wolstenholme's, Morley's and Lehmer's Congruence Theorems Revisited

Christian Aebi, Grant Cairns

Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems