2004/01/20 by Pingzhi Yuan · 3 citations
Physics and Astronomy · Mathematics · Computer Science · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Coding theory and cryptography
paper · pdf · doi:10.1090/s0002-9939-04-07418-0
In this paper, using a result of Ljunggren and some results on primitive prime factors of Lucas sequences of the first kind, we prove the following results by an elementary argument: if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are positive integers, then the simultaneous Pell equations <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x squared minus 4 m left-parenthesis m plus 1 right-parenthesis y squared equals y squared minus b z squared equals 1"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>4</mml:mn> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x2-4m(m+1)y2=y2-bz2=1</mml:annotation> </mml:semantics> </mml:math> </disp-formula> possesses at most one solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,y,z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in positive integers.