Type: Article
Publication Date: 2021-08-12
Citations: 9
DOI: https://doi.org/10.1155/2021/5379284
Polynomial Pell’s equation is <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>D</mi> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math> , where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>D</mi> </math> is a quadratic polynomial with integer coefficients and the solutions <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> </math> must be quadratic polynomials with integer coefficients. Let <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>D</mi> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> be a polynomial in <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi mathvariant="double-struck">Z</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> . In this paper, some quadratic polynomial solutions are given for the equation <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>D</mi> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math> which are significant from computational point of view.