We discuss, with the aid of arithmetical properties of the ring of the Gaussian integers, the solvability of the Diophantine equation <mml:math alttext="$ax^{2}+2bxy-4ay^{2}=\pm 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>b</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>4</mml:mn><mml:mi>a</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mo>&#177;</mml:mo><mml:mn>1</mml:mn></mml:math>, where <mml:math alttext="$a$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math> and <mml:math alttext="$b$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi></mml:math> are nonnegative integers. The discussion is relative to the solution of Pell&#39;s equation <mml:math alttext="$v^{2}-(4a^{2}+b^{2})w^{2}=-4$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>&#8722;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mo>&#8722;</mml:mo><mml:mn>4</mml:mn></mml:math>.

The Diophantine equation <mml:math alttext="$ax^{2}+2bxy-4ay^{2}=\pm 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>b</mml:mi><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mi>a</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:math>

The Diophantine equation <svg style="vertical-align:-3.56265pt" height="20.275" version="1.1" viewBox="0 0 165.60001 20.275" width="165.60001" 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="x1D44E" d="M483 97q-42 -50 -88.5 -79.5t-68.5 -29.5q-37 0 -17 93l22 102h-2q-54 -79 -144 -149q-59 -46 -100 -46q-24 0 -43 29t-19 86q0 78 34.5 153t94.5 120q41 31 94 51.5t98 20.5q29 0 72 -9q26 -6 39 -6l2 -4q-30 -117 -67 -323q-8 -41 2 -41q16 0 79 58zM374 387&#xA;q-32 15 -73 15q-52 0 -83 -23q-48 -36 -78 -108.5t-30 -152.5q0 -33 8.5 -50.5t20.5 -17.5q31 0 107 79t99 132q15 40 29 126z" /></g><g transform="matrix(.017,-0,0,-.017,8.664,15.775)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37&#xA;q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.012,-0,0,-.012,18.162,7.613)"><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><g transform="matrix(.017,-0,0,-.017,28.275,15.775)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,42.027,15.775)"><use xlink:href="#x32"/></g><g transform="matrix(.017,-0,0,-.017,50.186,15.775)"><path id="x1D44F" d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76&#xA;q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z" /></g><g transform="matrix(.017,-0,0,-.017,58.159,15.775)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,67.661,15.775)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26&#xA;q13 28 43 28q18 0 30 -15.5t12 -39.5z" /></g><g transform="matrix(.017,-0,0,-.017,81.277,15.775)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,95.029,15.775)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g><g transform="matrix(.017,-0,0,-.017,103.188,15.775)"><use xlink:href="#x1D44E"/></g><g transform="matrix(.017,-0,0,-.017,111.79,15.775)"><use xlink:href="#x1D466"/></g><g transform="matrix(.012,-0,0,-.012,121.65,7.613)"><use xlink:href="#x32"/></g><g transform="matrix(.017,-0,0,-.017,132.7,15.775)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,147.387,15.775)"><path id="xB1" d="M535 290h-212v-206h-58v206h-213v50h213v193h58v-193h212v-50zM535 -22h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,157.365,15.775)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g></svg> 

International Journal of Mathematics and Mathematical Sciences

The Diophantine equation ax2+2bxy−4ay2=±1

Abstract

Copyright