{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:11:23Z","timestamp":1776838283327,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"349","license":[{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this note we present a construction of an infinite family of diagonal quintic threefolds defined over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B equals left-parenthesis upper B 0 comma upper B 1 comma upper B 2 comma upper B 3 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B=(B_{0}, B_{1}, B_{2}, B_{3})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of co-prime integers such that for a suitable chosen integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (depending on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ), the equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B 0 upper X 0 Superscript 5 Baseline plus upper B 1 upper X 1 Superscript 5 Baseline plus upper B 2 upper X 2 Superscript 5 Baseline plus upper B 3 upper X 3 Superscript 5 Baseline equals b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:msubsup>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>5<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has infinitely many positive integer solutions.\n                  <\/p>","DOI":"10.1090\/mcom\/3953","type":"journal-article","created":{"date-parts":[[2024,2,7]],"date-time":"2024-02-07T11:43:22Z","timestamp":1707306202000},"page":"2503-2511","source":"Crossref","is-referenced-by-count":0,"title":["Construction of diagonal quintic threefolds with infinitely many rational points"],"prefix":"10.1090","volume":"93","author":[{"given":"Maciej","family":"Ulas","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,2,14]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"353","DOI":"10.2307\/2001512","article-title":"Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture","volume":"324","author":"Albano, Alberto","year":"1991","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"2","unstructured":"N. D. Elkies, The ABC\u2019s of number theory, Harvard Math. Rev. 1 (2007), 64\u201376."},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/BF01388432","article-title":"Endlichkeitss\u00e4tze f\u00fcr abelsche Variet\u00e4ten \u00fcber Zahlk\u00f6rpern","volume":"73","author":"Faltings, G.","year":"1983","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"1079","DOI":"10.1090\/S0002-9904-1966-11654-3","article-title":"Counterexample to Euler\u2019s conjecture on sums of like powers","volume":"72","author":"Lander, L. J.","year":"1966","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"1689","DOI":"10.4310\/pamq.2022.v18.n4.a12","article-title":"Singular plane sections and the conics in the Fermat quintic threefold","volume":"18","author":"Musta\u0163\u0103, Anca","year":"2022","journal-title":"Pure Appl. Math. Q.","ISSN":"https:\/\/id.crossref.org\/issn\/1558-8599","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"729","DOI":"10.2140\/pjm.2019.303.729","article-title":"Linearly dependent powers of binary quadratic forms","volume":"303","author":"Reznick, Bruce","year":"2019","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"key":"7","unstructured":"D. Testa, Conics on the Fermat quintic threefold, talk on the Junior Number Theory Seminar, October 11, 2010. Mathematical Insitute, Oxford. Avaliable at: \\url{https:\/\/math.columbia.edu\/ dejong\/reu\/lib\/exe\/fetch.php%3Fmedia=beamerjnts.pdf}"},{"key":"8","unstructured":"Wolfram Research, Inc., Mathematica, Version 13.2. 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