Pell's equation without irrational numbers

We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree.Comment: 10 pages, 3 figures added some references, fixed typos, added remarks on Speeding up the algorith

Solutions of polynomial Pell's equation

AbstractLet D=F2+2G be a monic quartic polynomial in Z[x], where degG<degF. Then for F/G∈Q[x], a necessary and sufficient condition for the solution of the polynomial Pell's equation X2−DY2=1 in Z[x] has been shown. Also, the polynomial Pell's equation X2−DY2=1 has nontrivial solutions X,Y∈Q[x] if and only if the values of period of the continued fraction of D are 2, 4, 6, 8, 10, 14, 18, and 22 has been shown. In this paper, for the period of the continued fraction of D is 4, we show that the polynomial Pell's equation has no nontrivial solutions X,Y∈Z[x]

Diophantine Equation

In the first chapter we have given some definations, theorems, lemmas on Elementary Number Theory. This brief discussion is useful for next discussion on the main topic. We know that there are two type of Diophantine equation i.e., (i) Linear Diophantine equation, (ii) Non-linear Diophantine equation. In the second chapter we have discussed about the solutions of both kind of Diophantine equation. Here we have given the necessary and sucient condition for existence the solution of a Linear Diophantine equation and also discussed about the Non-linear Diophantine equation and discussed Fermat's Last theorem. Pell's equation is a special type of Diophantine equation. The history of Pell's equation is very interesting. In the last section we have given some methods to find the fundamental solution of the Pell's equation

Generalized Pell's equations and Weber's class number problem

We study a generalization of Pell's equation, whose coefficients are certain algebraic integers. Let $X_0=0$ and $X_n=\sqrt{2+X_{n-1}}$ for each $n\in \mathbb{Z}_{\ge 1}$. We study the $\mathbb{Z}[X_{n-1}]$-solutions of the equation $x^2-X_n^2y^2=1$. By imitating the solution to the classical Pell's equation, we introduce new continued fraction expansions for $X_n$ over $\mathbb{Z}[X_{n-1}]$ and obtain an explicit solution of the generalized Pell's equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber's class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the $\mathbb{Z}_2$-extension over the rationals and show the convergence of the class numbers in $\mathbb{Z}_2$.Comment: 17 page

Pell's equation

Call number: LD2668 .R4 1966 N38

Pell's Equation

This article explains about Pell's number theory, a Diophantine equation, the number of variables is greater than the number of equations, allowing for the possibility of infinitely many solutions