Let <mml:math alttext="$L_n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> denote the <mml:math alttext="$n^{th} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math> Lucas number, where <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> is a natural number.
Using elementary techniques, we find all solutions of the equation: <mml:math alttext="$L_n  = px^2 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>L</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>
where <mml:math alttext="$p$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> is prime and <mml:math alttext="$p &#60; 1000$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>&#60;</mml:mo><mml:mn>1000</mml:mn></mml:math>.

Lucas numbers of the form <mml:math alttext="$PX^2 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi><mml:msup><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>, where <mml:math alttext="$P$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math> is prime

Lucas numbers of the form <svg style="vertical-align:-0.0pt" height="15.8375" version="1.1" viewBox="0 0 30.737499 15.8375" width="30.737499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D443" d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5&#xA;t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" /></g><g transform="matrix(.017,-0,0,-.017,10.874,15.775)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231&#xA;l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g><g transform="matrix(.012,-0,0,-.012,24.338,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg>, where <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 10.9375 11.175" width="10.9375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D443"/></g></svg> is prime

International Journal of Mathematics and Mathematical Sciences

Lucas numbers of the form PX2, where P is prime

Abstract

Copyright