{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:50:11Z","timestamp":1747191011798,"version":"3.40.5"},"reference-count":23,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T00:00:00Z","timestamp":1653955200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100019812","name":"Tishreen University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100019812","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2022,5,31]]},"abstract":"<jats:p>In this paper, we give a necessary and sufficient condition for real quadratic field <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <a:mi>k<\/a:mi>\n                        <a:mo>=<\/a:mo>\n                        <a:mi mathvariant=\"normal\">\u211a<\/a:mi>\n                        <a:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <a:mrow>\n                              <a:msqrt>\n                                 <a:mrow>\n                                    <a:mn>4<\/a:mn>\n                                    <a:mi>n<\/a:mi>\n                                    <a:msup>\n                                       <a:mrow>\n                                          <a:mtext>\u2009<\/a:mtext>\n                                       <\/a:mrow>\n                                       <a:mrow>\n                                          <a:mn>2<\/a:mn>\n                                       <\/a:mrow>\n                                    <\/a:msup>\n                                    <a:mo>+<\/a:mo>\n                                    <a:mn>1<\/a:mn>\n                                 <\/a:mrow>\n                              <\/a:msqrt>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                     <\/a:math>\n                  <\/jats:inline-formula> to have class number <jats:inline-formula>\n                     <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <g:mi>\u03c9<\/g:mi>\n                        <g:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <g:mrow>\n                              <g:mi>n<\/g:mi>\n                           <\/g:mrow>\n                        <\/g:mfenced>\n                        <g:mo>+<\/g:mo>\n                        <g:mi>c<\/g:mi>\n                     <\/g:math>\n                  <\/jats:inline-formula> where <jats:inline-formula>\n                     <l:math xmlns:l=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <l:mi>c<\/l:mi>\n                        <l:mo>\u2208<\/l:mo>\n                        <l:mfenced open=\"{\" close=\"}\" separators=\"|\">\n                           <l:mrow>\n                              <l:mn>1<\/l:mn>\n                              <l:mo>,<\/l:mo>\n                              <l:mtext>\u2009<\/l:mtext>\n                              <l:mn>2<\/l:mn>\n                           <\/l:mrow>\n                        <\/l:mfenced>\n                     <\/l:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <q:mi>\u03c9<\/q:mi>\n                        <q:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <q:mrow>\n                              <q:mi>n<\/q:mi>\n                           <\/q:mrow>\n                        <\/q:mfenced>\n                     <\/q:math>\n                  <\/jats:inline-formula> is the number of odd prime divisors of <jats:inline-formula>\n                     <v:math xmlns:v=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <v:mi>n<\/v:mi>\n                     <\/v:math>\n                  <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1155\/2022\/6579960","type":"journal-article","created":{"date-parts":[[2022,6,1]],"date-time":"2022-06-01T03:05:26Z","timestamp":1654052726000},"page":"1-4","source":"Crossref","is-referenced-by-count":0,"title":["A Necessary and Sufficient Condition for <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                     <a:mi>k<\/a:mi>\n                     <a:mo>=<\/a:mo>\n                     <a:mi mathvariant=\"normal\">\u211a<\/a:mi>\n                     <a:mfenced open=\"(\" close=\")\" separators=\"|\">\n                        <a:mrow>\n                           <a:msqrt>\n                              <a:mrow>\n                                 <a:mn>4<\/a:mn>\n                                 <a:mi>n<\/a:mi>\n                                 <a:msup>\n                                    <a:mrow>\n                                       <a:mtext>\u2009<\/a:mtext>\n                                    <\/a:mrow>\n                                    <a:mrow>\n                                       <a:mn>2<\/a:mn>\n                                    <\/a:mrow>\n                                 <\/a:msup>\n                                 <a:mo>+<\/a:mo>\n                                 <a:mn>1<\/a:mn>\n                              <\/a:mrow>\n                           <\/a:msqrt>\n                        <\/a:mrow>\n                     <\/a:mfenced>\n                  <\/a:math> to Have Class Number <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                     <g:mi>\u03c9<\/g:mi>\n                     <g:mfenced open=\"(\" close=\")\" separators=\"|\">\n                        <g:mrow>\n                           <g:mi mathvariant=\"bold\">n<\/g:mi>\n                        <\/g:mrow>\n                     <\/g:mfenced>\n                     <g:mo>+<\/g:mo>\n                     <g:mi mathvariant=\"bold\">c<\/g:mi>\n                  <\/g:math>"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4768-0034","authenticated-orcid":true,"given":"Ahmad","family":"Issa","sequence":"first","affiliation":[{"name":"Department of Mathematics, Tishreen University, Lattakia, Syria"}]},{"given":"Hasan","family":"Sankari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tishreen University, Lattakia, Syria"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4939-7560-0","volume-title":"Disquisitiones Arithmeticae","author":"C. 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