{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:13:04Z","timestamp":1776831184225,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"244","license":[{"start":{"date-parts":[[2004,5,1]],"date-time":"2004-05-01T00:00:00Z","timestamp":1083369600000},"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                    We present a computational approach for finding all integral solutions of the equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y squared equals 1 Superscript k Baseline plus 2 Superscript k Baseline plus midline-horizontal-ellipsis plus x Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">y^2=1^k+2^k+\\dotsb +x^k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for even values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 less-than-or-equal-to k less-than-or-equal-to 70\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>70<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2\\le k\\le 70<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    assuming the Generalized Riemann Hypothesis, and for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 less-than-or-equal-to k less-than-or-equal-to 58\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>58<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2\\le k\\le 58<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    unconditionally.\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01465-0","type":"journal-article","created":{"date-parts":[[2003,6,20]],"date-time":"2003-06-20T10:23:27Z","timestamp":1056104607000},"page":"2099-2110","source":"Crossref","is-referenced-by-count":15,"title":["A computational approach for solving \ud835\udc66\u00b2=1^{\ud835\udc58}+2^{\ud835\udc58}+\u2026+\ud835\udc65^{\ud835\udc58}"],"prefix":"10.1090","volume":"72","author":[{"suffix":"Jr.","given":"M.","family":"Jacobson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u00c1.","family":"Pint\u00e9r","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P.","family":"Walsh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,5,1]]},"reference":[{"issue":"12","key":"1","doi-asserted-by":"publisher","first-page":"3481","DOI":"10.1090\/S0002-9939-99-05041-8","article-title":"The Diophantine equation \ud835\udc4f\u00b2\ud835\udc4b\u2074-\ud835\udc51\ud835\udc4c\u00b2=1","volume":"127","author":"Bennett, Michael A.","year":"1999","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"99","DOI":"10.4064\/aa-44-2-99-107","article-title":"On some generalizations of the Diophantine equation 1^{\ud835\udc58}+2^{\ud835\udc58}+\u22ef+\ud835\udc65^{\ud835\udc58}=\ud835\udc66^{\ud835\udc67}","volume":"44","author":"Brindza, B.","year":"1984","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"3","isbn-type":"print","first-page":"595","article-title":"Power values of sums 1^{\ud835\udc58}+2^{\ud835\udc58}+\u22ef+\ud835\udc65^{\ud835\udc58}","author":"Brindza, B.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0444704752"},{"issue":"3-4","key":"4","doi-asserted-by":"publisher","first-page":"271","DOI":"10.5486\/pmd.2000.2354","article-title":"On the number of solutions of the equation 1^{\ud835\udc58}+2^{\ud835\udc58}+\u22ef+(\ud835\udc65-1)^{\ud835\udc58}=\ud835\udc66^{\ud835\udc67}","volume":"56","author":"Brindza, B.","year":"2000","journal-title":"Publ. Math. Debrecen","ISSN":"https:\/\/id.crossref.org\/issn\/0033-3883","issn-type":"print"},{"key":"5","isbn-type":"print","first-page":"159","article-title":"Short representation of quadratic integers","author":"Buchmann, Johannes","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0792335015"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"401","DOI":"10.4064\/aa-78-4-401-403","article-title":"The Diophantine equation \ud835\udc65\u2074-\ud835\udc37\ud835\udc66\u00b2=1. II","volume":"78","author":"Cohn, J. H. E.","year":"1997","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"7","first-page":"383","article-title":"On a Diophantine equation involving quadratic characters","volume":"57","author":"Dilcher, Karl","year":"1986","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"386","key":"8","doi-asserted-by":"publisher","first-page":"iv+94","DOI":"10.1090\/memo\/0386","article-title":"Zeros of Bernoulli, generalized Bernoulli and Euler polynomials","volume":"73","author":"Dilcher, Karl","year":"1988","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"171","DOI":"10.4064\/aa-68-2-171-192","article-title":"Computing integral points on elliptic curves","volume":"68","author":"Gebel, J.","year":"1994","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"521","DOI":"10.2307\/1968936","article-title":"A class of orthogonal functions on plane curves","volume":"40","author":"Jackson, Dunham","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"11","unstructured":"M. J. Jacobson, Jr., Subexponential Class Group Computation in Quadratic Orders, Ph.D. thesis, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany, 1999."},{"issue":"3","key":"12","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1080\/10586458.1995.10504322","article-title":"An investigation of bounds for the regulator of quadratic fields","volume":"4","author":"Jacobson, Michael J., Jr.","year":"1995","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"4","key":"13","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1080\/10586458.2000.10504666","article-title":"The size of the fundamental solutions of consecutive Pell equations","volume":"9","author":"Jacobson, Michael J., Jr.","year":"2000","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"key":"14","unstructured":"M. J. Jacobson, Jr. and H. C. Williams, Modular arithmetic on elements of small norm in quadratic fields, Submitted to Designs, Codes, and Cryptography."},{"key":"15","doi-asserted-by":"crossref","unstructured":"D. H. Lehmer, An extended theory of Lucas functions, Ann. Math. 31 (1930), 419\u2013448.","DOI":"10.2307\/1968235"},{"key":"16","isbn-type":"print","first-page":"123","article-title":"On the calculation of regulators and class numbers of quadratic fields","author":"Lenstra, H. W., Jr.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0521285135"},{"key":"17","unstructured":"The LiDIA Group, LiDia: a C++ library for computational number theory, Software, Technische Univesit\u00e4t Darmstadt, Germany, 1997, see http:\/\/www.informatik.tu-darmstadt.de\/TI\/LiDIA."},{"key":"18","unstructured":"\u00c9. Lucas, Solution de la question 1180, Nouv. Ann. Math. (2) 16 (1877), 429\u2013432."},{"key":"19","unstructured":"\u00c1. Pint\u00e9r, On a conjecture of Schaffer concerning the power values of power sums, preprint, (2000)."},{"key":"20","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/1989990","article-title":"Steinitz field towers for modular fields","volume":"46","author":"MacLane, Saunders","year":"1939","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"21","series-title":"Cambridge Tracts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511566042","volume-title":"Exponential Diophantine equations","volume":"87","author":"Shorey, T. N.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0521268265"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"349","DOI":"10.4064\/aa-51-4-349-368","article-title":"On the equation \ud835\udc53(1)1^{\ud835\udc58}+\ud835\udc53(2)2^{\ud835\udc58}+\u22ef+\ud835\udc53(\ud835\udc65)\ud835\udc65^{\ud835\udc58}+\ud835\udc45(\ud835\udc65)=\ud835\udc4f\ud835\udc66^{\ud835\udc67}","volume":"51","author":"Urbanowicz, Jerzy","year":"1988","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"23","series-title":"Canadian Mathematical Society Series of Monographs and Advanced Texts","isbn-type":"print","volume-title":"\\'{E}douard Lucas and primality testing","volume":"22","author":"Williams, Hugh C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0471148520"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01465-0\/S0025-5718-03-01465-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01465-0\/S0025-5718-03-01465-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:26:01Z","timestamp":1776727561000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01465-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,5,1]]},"references-count":23,"journal-issue":{"issue":"244","published-print":{"date-parts":[[2003,10]]}},"alternative-id":["S0025-5718-03-01465-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01465-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2003,5,1]]}}}