Related papers: Pell's equation without irrational numbers
In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell's equation for $D$. It is well-known that, once an integer solution to Pell's equation…
Using the representation of numbers, the methods of decreasing the number of calculation steps for Pell's equation are developed. The formulae relating the natural solution x0 of Pell's equation for A0 with the infinite number of minimal…
The purpose of this article is to give the solutions of the inverse problem for Pellian equations. For any rational number $0< a/b < 1$, the fundamental discriminants $D$ satisfying $(\lfloor \sqrt{D} \rfloor b + a)^2 - D b^2 = 4$ are given…
A proof of Bell's theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.
It is important in drawing techniques to find combinations of two straight lines and their angle bisectors whose slopes are all rational numbers. This problem is reduced to solving the Diophantine equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$…
For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions…
We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant.…
We present a method that maintains a balanced binary search tree without using any tree balance criterion at all, with the ultimate aim of maximum simplicity. In fact, our method is highly intuitive, and we only need to add minimal extra…
We introduce a generalization of the continued fraction and Chakravala algorithms for solving the Pell equation, utilizing the LLL-algorithm for rank 2 lattices.
The classical Pell equation can be extended to the cubic case considering the elements of norm one in $Z[\sqrt[3]{r}]$, which satisfy $x^3 + r y^3 + r^2 z^3 - 3 r x y z = 1$. The solution of the cubic Pell equation is harder than the…
We revisit the problem of rational search: given an unknown rational number $\alpha = \frac{a}{b} \in (0,1)$ with $b \leq n$, the goal is to identify $\alpha$ using comparison queries of the form ``$\beta \leq \alpha$?''. The problem has…
Cabello proved Bell's theorem without using inequalities. A loophole of Cabello's proof is pointed out in this work.
Most of known multipartite Bell inequalities involve correlation functions for all subsystems. They are useless for entangled states without such correlations. We give a method of derivation of families of Bell inequalities for N parties,…
A simple approach is presented to study the asymptotic behavior of some algorithms with an underlying tree structure. It is shown that some asymptotic oscillating behaviors can be precisely analyzed without resorting to complex analysis…
We present an empirical-yet-rigorous approach for solving a wide class of functional equations, thereby automating many results that previously required considerable human ingenuity and human labor.
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
In this paper, we prove that there is no number with the Lehmer property in the sequence of Pell numbers.
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some…