Related papers: Pythagorean Triangles with Repeated Digits-Repeate…

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group…

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in…

Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…

In 1966, Alan Sutcliffe published an introductory paper on "numbers which are multiplied when their digits are reversed". Two years later T. J. Kaczynski extended Sutcliffe's work to show that there exists a 3 digit number in base n with…

Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle…

A set of m distinct positive integers {a_{1},...a_{m}} is called a Diophantine m-tuple if a_{i}a_{j}+n is a square for each 1\leqi<j\leqm . The aim of this study is to show that some P_{k} sets can not be extendible to a Diophantine…

Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…

A regular truncated pyramid with rectangular bases;consists of two rectangular bases whose centers are orthogonally aligned with respect to the parallel planes containing their bases; and two pairs of congruent isosceles trapezoids(the four…

A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore,…

In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti ratios $A/a, B/b$. In particular, we prove that there are infinitely many essentially different ("non-similar")…

In 1987, Orrin Frink introduced the concept of almost Pythagorean triples. He defined them as an ordered triple $(x,y,z)$ that satisfies the equation $x^2+y^2=z^2+1$ where $x,y$ and $z$ are positive integers. In his paper, he showed that…

In a Note added in proof to a 1984 paper, Daniel A. Marcus claimed to have a counterexample to his conjecture that a minimal positively k-spanning vector configuration in R^m has size at most 2km. However, the counterexample was never…

For a positive integer $k$, we say that a graph is $k$-existentially complete if for every $0 \leq a \leq k$, and every tuple of distinct vertices $x_1,\ldots,x_a$, $y_1,\ldots,y_{k-a}$, there exists a vertex $z$ that is joined to all of…

It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately,…

A set of $m$ positive integers $\{x_{1},\ldots,x_{m}\}$ is called a $P^{3}_{1}$-set of size $m$ if the product of any three elements in the set increased by one is a cube integer. A $P^{3}_{1}$-set $S$ is said to be extendible if there…

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, \'E. Borel suggested that the answer is no and that for any real irrational algebraic number $x$…

A Pythagorean triple is a triple of positive integers a, b, c $\in$ N${}^{+}$ satisfying a${}^2$ + b${}^2$ = c${}^2$. Is it true that, for any finite coloring of N${}^{+}$ , at least one Pythagorean triple must be monochromatic? In other…

A d-dimensional simplex S is called a k-reptile if it can be tiled without overlaps by simplices S_1,S_2,...,S_k that are all congruent and similar to S. For d=2, k-reptile simplices (triangles) exist for many values of k and they have been…

We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7},$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least $6$. We will also provide an upper…

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be…