Related papers: Congruent numbers on the right trapezoid
From Euclid's fundamental formula for the Pythagorean triples we define the rational triples relating certain congruent numbers by an identity and explore their relationships. We introduce two geometric methods relating the congruent number…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
In this paper we present many congruences for several Ap\'ery-like sequences.
We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$. The asymptotic growth of this series depends explicitly on whether or not $t$ is a \emph{congruent number}, an integer that is the area of a…
We show the nonequivalence of combinations of several natural geometric restrictions on trapezoid representations of trapezoid orders. Each of the properties unit parallelogram, unit trapezoid and proper parallelogram, unit trapezoid and…
In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…
The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well known. We show how this correspondence can be extended to the…
We prove several congruences for trinomial coefficients.
In this paper, we extend the work of \cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\tau$-congruent numbers generated by them. We then characterize…
Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers.
We characterize the order of principal congruences of a bounded lattice as a bounded ordered set. We also state a number of open problems in this new field.
We prove some symmetric $q$-congruences.
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a…
The notion of $\theta$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $\theta$-congruent if it is the area of a rational triangle with an angle $\theta$ whose cosine is rational.…
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.