{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:11:47Z","timestamp":1776766307753,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"227","license":[{"start":{"date-parts":[[2000,2,24]],"date-time":"2000-02-24T00:00:00Z","timestamp":951350400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We obtain nonexistence conditions of a solution for of the congruence\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x 1 Superscript k Baseline plus midline-horizontal-ellipsis plus x Subscript s Superscript k Baseline identical-to upper N left-parenthesis mod p Superscript n Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mspace width=\"0.667em\"\/>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>mod<\/mml:mi>\n                            <mml:mspace width=\"0.333em\"\/>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x_1^k+\\cdots +x_s^k\\equiv N\\pmod {p^n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">s\\geq 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are integers, and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Superscript n\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">p^n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a prime power. We give nonexistence conditions of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis s comma upper N mod p Superscript n Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo lspace=\"thickmathspace\" rspace=\"thickmathspace\">mod<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msup>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(s, N\\bmod {p^n})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4\">\n                        <mml:semantics>\n                          <mml:mn>4<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5\">\n                        <mml:semantics>\n                          <mml:mn>5<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7\">\n                        <mml:semantics>\n                          <mml:mn>7<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis s comma p Superscript n Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(s, p^n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k equals 11\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>11<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=11<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"13\">\n                        <mml:semantics>\n                          <mml:mn>13<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">13<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"17\">\n                        <mml:semantics>\n                          <mml:mn>17<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">17<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"19\">\n                        <mml:semantics>\n                          <mml:mn>19<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">19<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Furthermore, we complete some tables concerned with Waring\u2019s problem in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -adic fields that were computed by Hardy and Littlewood.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01067-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1283-1297","source":"Crossref","is-referenced-by-count":2,"title":["Nonexistence conditions of a solution for the congruence \ud835\udc65\u2081^{\ud835\udc58}+\u22ef+\ud835\udc65_{\ud835\udc60}^{\ud835\udc58}\u2261\ud835\udc41(\\mod\ud835\udc5d\u207f)"],"prefix":"10.1090","volume":"68","author":[{"given":"Hiroshi","family":"Sekigawa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kenji","family":"Koyama","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,2,24]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"169","DOI":"10.4064\/aa-9-2-169-176","article-title":"Waring\u2019s problem for \ud835\udd2d-adic number fields","volume":"9","author":"Birch, B. J.","year":"1964","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"343","DOI":"10.4064\/aa-29-4-343-351","article-title":"A note on Waring\u2019s problem in \ud835\udc5d-fields","volume":"29","author":"Bovey, J. D.","year":"1976","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"315","DOI":"10.4064\/aa-22-3-315-327","article-title":"On Waring\u2019s problem in \ud835\udc5d-adic fields","volume":"22","author":"Dodson, M. M.","year":"1972","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"G. H. Hardy and J. E. Littlewood, Some problems of \u2018Partitio Numerorum\u2019 (VIII): The number \u0393(\ud835\udc58) in Waring\u2019s problem, Proc. London Math. Soc. 28 (1928), 518\u2013542.","DOI":"10.1112\/plms\/s2-28.1.518"},{"issue":"218","key":"5","doi-asserted-by":"publisher","first-page":"841","DOI":"10.1090\/S0025-5718-97-00830-2","article-title":"On searching for solutions of the Diophantine equation \ud835\udc65\u00b3+\ud835\udc66\u00b3+\ud835\udc67\u00b3=\ud835\udc5b","volume":"66","author":"Koyama, Kenji","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","series-title":"Pure and Applied Mathematics, Vol. 30","volume-title":"Diophantine equations","author":"Mordell, L. J.","year":"1969"},{"key":"7","doi-asserted-by":"publisher","first-page":"783","DOI":"10.2307\/2371336","article-title":"Ring homomorphisms which are also lattice homomorphisms","volume":"61","author":"Ward, Morgan","year":"1939","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"783","DOI":"10.2307\/2371336","article-title":"Ring homomorphisms which are also lattice homomorphisms","volume":"61","author":"Ward, Morgan","year":"1939","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01067-4\/S0025-5718-99-01067-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01067-4\/S0025-5718-99-01067-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:06:11Z","timestamp":1776722771000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01067-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,24]]},"references-count":8,"journal-issue":{"issue":"227","published-print":{"date-parts":[[1999,7]]}},"alternative-id":["S0025-5718-99-01067-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01067-4","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":[[1999,2,24]]}}}