On the x-coordinates of Pell equations that are sums of two Padovan numbers

TitleOn the x-coordinates of Pell equations that are sums of two Padovan numbers
Publication TypeJournal Article
Year of Publication2021
AuthorsDdamulira, M
Volume27
Issue4
Date Published23 February 2021
Abstract

Let (</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi></mrow></msub><msub><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>&#x2265;</mo><mn>0</mn></mrow></msub></math>" id="MathJax-Element-1-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">(Pn)n≥0(Pn)n≥0 be the sequence of Padovan numbers defined by <mi>P</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></math>" id="MathJax-Element-2-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">P0=0P0=0, <mi>P</mi><mn>1</mn></msub><mo>=</mo><msub><mi>P</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn></math>" id="MathJax-Element-3-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">P1=P2=1P1=P2=1, and <mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>3</mn></mrow></msub><mo>=</mo><msub><mi>P</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>P</mi><mi>n</mi></msub></math>" id="MathJax-Element-4-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">Pn+3=Pn+1+PnPn+3=Pn+1+Pn for all n</mi><mo>&#x2265;</mo><mn>0</mn></math>" id="MathJax-Element-5-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">n≥0n≥0. In this paper, we find all positive square-free integers d such that the Pell equations <mi>x</mi><mn>2</mn></msup><mo>&#x2212;</mo><mi>d</mi><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>N</mi></math>" id="MathJax-Element-6-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">x2−dy2=Nx2−dy2=N with N</mi><mo>&#x2208;</mo><mo fence="false" stretchy="false">{</mo><mo>&#x00B1;</mo><mn>1</mn><mo>,</mo><mo>&#x00B1;</mo><mn>4</mn><mo fence="false" stretchy="false">}</mo></math>" id="MathJax-Element-7-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">N∈{±1,±4}N∈{±1,±4}, have at least two positive integer solutions (x, y) and (</mo><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msup><mo>,</mo><msup><mi>y</mi><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msup><mo stretchy="false">)</mo></math>" id="MathJax-Element-8-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">(x′,y′)(x′,y′) such that both x and <mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi class="MJX-variant" mathvariant="normal">&#x2032;</mi></mrow></msup></math>" id="MathJax-Element-9-Frame" role="presentation" style="box-sizing: inherit; display: inline; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">x′x′ are sums of two Padovan numbers.

URLhttps://link.springer.com/article/10.1007/s40590-021-00312-8