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We give p-adic interpretation of this equation.","publication_date":"2011,12,27"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"A p-adic look at the Diophantine equation x^2+11^2k=y^n","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [146832572]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":84954988,"attachmentType":"pdf"}"><img alt="First page of “A p-adic look at the Diophantine equation x^2+11^2k=y^n”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/84954988/mini_magick20220426-11154-1vh8dj7.png?1650959174" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">A p-adic look at the Diophantine equation x^2+11^2k=y^n</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="146832572" href="https://independent.academia.edu/YilmazSimsek2"><img alt="Profile image of Yilmaz Simsek" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Yilmaz Simsek</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2011</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">4 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 77100435; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=77100435"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { const viewCountNumber = Number(viewCount); if (!viewCountNumber) { throw new Error('Failed to parse view count'); } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (viewCountBody) { viewCountBody.textContent = `${commaizedViewCount} views`; } else { throw new Error('Failed to find work views element'); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We find all solutions of Diophantine equation x^2+11^2k = y^n where x&gt;=1, y&gt;=1, n&gt;=3 and k is natural number. We give p-adic interpretation of this equation.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":84954988,"attachmentType":"pdf","workUrl":"https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":84954988,"attachmentType":"pdf","workUrl":"https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="84954988" data-landing_url="https://www.academia.edu/77100435/A_p_adic_look_at_the_Diophantine_equation_x_2_11_2k_y_n" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="24497746" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/24497746/A_p_adic_Look_at_the_Diophantine_Equation_x_sup_2_11_sup_2k_y_sup_n_">A p-adic Look at the Diophantine Equation x[sup 2]+11[sup 2k] = y[sup n]</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="47214595" 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href="https://www.academia.edu/2908634/On_the_diophantine_equation_x_2_5_a_cdot_11_b_y_n_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="32615017" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/32615017/A_study_of_Fermats_Last_Theorem_and_other_Diophantine_Equations">A study of Fermat's Last Theorem and other Diophantine Equations</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="63277192" href="https://oauife.academia.edu/OlufemiOYADARE">Olufemi OYADARE</a></div><p class="ds-related-work--abstract ds2-5-body-sm">This paper develops a framework of algebra whereby every Diophan-tine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Binomial triangle. We then apply the framework to the discussion of some notable results in the theory of numbers. Among other results, we prove a new and complete generation of all Pythagorean triples (without necessarily resorting to their production by examples), convert the collection of Bi-nomial triangles to a Noetherian ring (whose identity element is found to be the well-known Pascal triangle) and develop an easy understanding of the original Fermat's Last Theorem (F LT). The application includes the computation of the Galois groups of those polynomials coming from our outlook on F LT and an approach to the explicit realization of arithmetic groups of curves by a treatment of some Diophantine curves.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A study of Fermat's Last Theorem and other Diophantine Equations","attachmentId":52787831,"attachmentType":"pdf","work_url":"https://www.academia.edu/32615017/A_study_of_Fermats_Last_Theorem_and_other_Diophantine_Equations","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/32615017/A_study_of_Fermats_Last_Theorem_and_other_Diophantine_Equations"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="15345081" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/15345081/On_the_Diophantine_equation_x2_2a11b_yn">On the Diophantine equation x2+ 2a11b= yn</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="34464840" href="https://unideb.academia.edu/%C3%81kosPint%C3%A9r">Ákos Pintér</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2010</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline 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