STRUCTURE OF SOLUTIONS TO THE PELL EQUATION

​ABSTRACT

In the section of algebra, a significant role is played by the study of all types of equations and methods for solving them, and the emphasis is placed on them, a large amount of time is allocated. It is thanks to them that the most important tasks related to the cognition of reality are written in symbolic language. This article considers the structure of solutions to the Pell equation.

Keywords: pell's equation, linear diophantine equations, continued fractions.

The classical operations in mathematics include the solution of elementary expressions with several unknowns. They are called linear Diophantine equations. The theory developed by the ancient Greek scientist allows you to calculate equalities without using complex formulas. The method is based on reasoning and a clear understanding of the number theory, connected in a logical construction. At school, it is studied in the eighth grade, and it is also widely used in practice. The search for ways to solve linear equations began in ancient Babylon and Greece. The ancient Greek philosopher and mathematician of the ruler of Greece, Diophantus of Alexandria, was able to achieve tremendous success in solving linear equations [3].

The classical operations in mathematics include the solution of elementary expressions with several unknowns. They are called linear Diophantine equations, and Pell's equation belongs to them -  - m =1 [1].

In order to solve the equation, there are many ways, but not one answers the question "is it possible that the solution does not exist in principle, and if it does exist, then how to find it?".

If the Pell equation has at least one non-trivial solution, then multiplying it many times by itself, one can find infinitely many solutions [5,6].

Theorem 1. The Pell equation has a non-trivial solution[4].

Proof:

Suppose . For any natural n>1, by virtue of the lemma, there exist such natural numbers   and . that  < n and <. Further

=. Therefore, it takes a finite number of values. But n takes on an infinite number of values. Therefore, there exists a number c such that there are infinitely many pairs for it (, ), such that .

Consider the remainders after division by  numbers , . The number of remainders is finite and the number of pairs is infinite, so two distinguishable pairs , ) and , ), such that .

==

==

Because the -, then the numbers

 x =  and y= whole

.  =   =  = c=1.

That's why (x,y) - the desired nontrivial solution of the Pell equation.

Theorem 2 Any solution of the Pell equation is a convergent for −.

Proof:

 We consider x, y >0, the remaining roots are obtained from symmetry. Because   1, that

 x > y> 0. x + y > 2y. Hence 1= - d = (x -y) (x +y)> (x - y) 2y. Divide both parts by 2 and get:  -  < . Hence, by the approximation theorem  is an convergent for  

There is a way to solve Pell's equation using continued fractions. Their role in the theory of Pell's equations is no less significant than the role of irrational numbers. And the continued fractions themselves are extremely interesting.

Theorem 3. If an irreducible fraction     is such that  , , then   is a convergent of a number.

Theorem 4. Let (x, y) be a positive solution to the Pell equation. Then  is a suitable fraction .

Proof: Since х>у>0 and >1, that х+у>2у.

Means, 1=х² – mу² = (х - у)(х+у)>(х - у)· 2у.

We divide the resulting inequality by 2у² : . Since (x, y) is a positive solution of Pell's equation, the left side of this inequality is positive and the fraction   is irreducible. Therefore, according to Theorem 3, it is a convergent of the number   .

So, positive solutions of Pell's equations should be sought only among pairs made up of the numerator and denominator of some suitable fraction of the number  . The question arises as to which convergents correspond to the solutions of Pell's equation. The answer to it is given by a theorem, which is given without proof.

References:

1. Bugaenko, V.O. Pell's equations. 2nd ed., rev. and additional - M.: Publishing House of the Moscow Center for Continuous Mathematical Education, 2010. -36 p.
2. Bruno, A.D. Expanding algebraic numbers into continued fractions // Zhurnal vychislitel'noi matem. and mat. Physics. -1964, - N 2. - S. 211-221.
3. Van. Der Warden, B. L. Awakening Science: The Mathematics of Ancient Egypt, Babylon, and Greece. – M.: Fizmatgiz, 1959. – 460 p.
4. Zatorsky, R.A. On rational approximations of higher-order algebraic numbers and some parametrization of generalized Pell's equations. / Carpathian National University. V. Stefanik Ukraine, 2011. Access mode: https://arxiv.org/pdf/1103.5772.pdf (Accessed 20.11.2021).
5. Zverkina, G. A. The Pell-Fermat Equation in Antiquity. International scientific conference "Education, science and economics in universities at the turn of the millennium". Digest of articles. Vysoke Tatry, - 2000. - C. 210-213.
6. Edwards, G. Fermat's Last Theorem. Genetic introduction to algebraic number theory. - M.: Mir, 1980. - 477 p.