涉及三互易数特殊和的同余式
IF 1.3
4区 数学
Q1 MATHEMATICS
Zhongyan Shen
{"title":"涉及三互易数特殊和的同余式","authors":"Zhongyan Shen","doi":"10.1155/2024/8445635","DOIUrl":null,"url":null,"abstract":"Define the sums of triple reciprocals <span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.68632 35.781 15.5493\" width=\"35.781pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,8.996,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,13.494,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,20.02,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,28.15,0)\"></path></g></svg><span></span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"39.363183799999995 -9.68632 54.435 15.5493\" width=\"54.435pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,39.413,.007)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,49.2,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,51.748,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,57.308,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,61.222,3.466)\"><use xlink:href=\"#g54-36\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,66.782,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,71.414,3.466)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,76.974,3.466)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,82.252,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,88.492,0)\"></path></g></svg><span></span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"93.8031838 -9.68632 17.77 15.5493\" width=\"17.77pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,93.853,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,96.358,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,102.119,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,108.659,0)\"></path></g></svg><span></span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"113.75218380000001 -9.68632 6.513 15.5493\" width=\"6.513pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,113.802,0)\"><use xlink:href=\"#g113-106\"></use></g><g transform=\"matrix(.013,0,0,-0.013,117.351,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"122.4441838 -9.68632 8.464 15.5493\" width=\"8.464pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,122.494,0)\"><use xlink:href=\"#g113-107\"></use></g><g transform=\"matrix(.013,0,0,-0.013,127.994,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"133.0871838 -9.68632 17.802 15.5493\" width=\"17.802pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,133.137,0)\"><use xlink:href=\"#g113-108\"></use></g><g transform=\"matrix(.013,0,0,-0.013,143.308,0)\"></path></g></svg><span></span><span><svg height=\"15.5493pt\" style=\"vertical-align:-5.86298pt\" version=\"1.1\" viewbox=\"154.5211838 -9.68632 6.697 15.5493\" width=\"6.697pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,154.571,0)\"><use xlink:href=\"#g113-50\"></use></g></svg>.</span></span> Zhao discovered the following curious congruence for any odd prime <span><svg height=\"10.2124pt\" style=\"vertical-align:-3.42943pt\" version=\"1.1\" viewbox=\"-0.0498162 -6.78297 7.83752 10.2124\" width=\"7.83752pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"></path></g></svg>,</span> <span><svg height=\"14.7625pt\" style=\"vertical-align:-5.47417pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 36.965 14.7625\" width=\"36.965pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.013,0,0,-0.013,8.996,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.494,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,21.204,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,29.334,0)\"></path></g></svg><span></span><svg height=\"14.7625pt\" style=\"vertical-align:-5.47417pt\" version=\"1.1\" viewbox=\"40.5471838 -9.28833 86.298 14.7625\" width=\"86.298pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,40.597,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,48.228,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,54.468,0)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,62.216,3.132)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,67.65,3.132)\"></path></g><g transform=\"matrix(.0091,0,0,-0.0091,73.21,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,78.197,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,82.695,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,93.289,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,100.037,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,111.346,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,119.056,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,123.554,0)\"></path></g></svg><span></span></span> Xia and Cai extended the above congruence to modulo <span><svg height=\"17.0656pt\" style=\"vertical-align:-5.474199pt\" version=\"1.1\" viewbox=\"-0.0498162 -11.5914 15.621 17.0656\" width=\"15.621pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.71,-5.741)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,12.657,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"17.0656pt\" style=\"vertical-align:-5.474199pt\" version=\"1.1\" viewbox=\"17.750183800000002 -11.5914 36.965 17.0656\" width=\"36.965pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,17.8,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.013,0,0,-0.013,26.796,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,31.294,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,39.004,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,47.134,0)\"><use xlink:href=\"#g117-35\"></use></g></svg><span></span><svg height=\"17.0656pt\" style=\"vertical-align:-5.474199pt\" version=\"1.1\" viewbox=\"58.347183799999996 -11.5914 41.564 17.0656\" width=\"41.564pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,58.397,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,64.637,0)\"><use xlink:href=\"#g113-51\"></use></g><g transform=\"matrix(.013,0,0,-0.013,70.877,0)\"><use xlink:href=\"#g113-67\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,78.625,3.132)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,84.058,3.132)\"><use xlink:href=\"#g54-33\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,89.619,3.132)\"><use xlink:href=\"#g50-52\"></use></g><g transform=\"matrix(.013,0,0,-0.013,94.605,0)\"><use xlink:href=\"#g113-48\"></use></g></svg><span></span><svg height=\"17.0656pt\" style=\"vertical-align:-5.474199pt\" version=\"1.1\" viewbox=\"99.9161838 -11.5914 89.604 17.0656\" width=\"89.604pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,99.966,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,104.464,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,115.08,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,125.617,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,131.857,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,139.26,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,149.797,0)\"><use xlink:href=\"#g113-52\"></use></g><g transform=\"matrix(.013,0,0,-0.013,156.037,0)\"><use xlink:href=\"#g113-67\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,163.785,3.132)\"><use xlink:href=\"#g50-51\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,168.217,3.132)\"><use xlink:href=\"#g50-113\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,173.65,3.132)\"><use xlink:href=\"#g54-33\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,179.21,3.132)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,184.214,0)\"><use xlink:href=\"#g113-48\"></use></g></svg><span></span><svg height=\"17.0656pt\" style=\"vertical-align:-5.474199pt\" version=\"1.1\" viewbox=\"189.52518379999998 -11.5914 90.404 17.0656\" width=\"90.404pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,189.575,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,194.073,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,204.689,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,215.225,0)\"><use xlink:href=\"#g113-51\"></use></g><g transform=\"matrix(.013,0,0,-0.013,221.465,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,225.963,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,230.461,0)\"><use xlink:href=\"#g121-107\"></use></g><g transform=\"matrix(.013,0,0,-0.013,241.056,0)\"><use xlink:href=\"#g121-109\"></use></g><g transform=\"matrix(.013,0,0,-0.013,247.803,0)\"><use xlink:href=\"#g121-98\"></use></g><g transform=\"matrix(.013,0,0,-0.013,259.112,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,266.822,-5.741)\"><use xlink:href=\"#g50-51\"></use></g><g transform=\"matrix(.013,0,0,-0.013,271.769,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,276.267,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span></span> where <span><svg height=\"11.5434pt\" style=\"vertical-align:-3.42945pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.11395 18.973 11.5434\" width=\"18.973pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,11.342,0)\"></path></g></svg><span></span><svg height=\"11.5434pt\" style=\"vertical-align:-3.42945pt\" version=\"1.1\" viewbox=\"22.555183800000002 -8.11395 6.419 11.5434\" width=\"6.419pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,22.605,0)\"></path></g></svg></span> is a prime. In this paper, we consider the congruences about <span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 60.406 12.7178\" width=\"60.406pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-91\"></use></g><g transform=\"matrix(.013,0,0,-0.013,8.996,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,13.494,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,17.979,0)\"><use xlink:href=\"#g113-41\"></use></g><g transform=\"matrix(.013,0,0,-0.013,22.477,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.013,0,0,-0.013,33.093,0)\"><use xlink:href=\"#g117-33\"></use></g><g transform=\"matrix(.013,0,0,-0.013,43.63,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,52.775,0)\"></path></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"63.2611838 -9.28833 20.765 12.7178\" width=\"20.765pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,63.311,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,74.222,0)\"><use xlink:href=\"#g113-42\"></use></g><g transform=\"matrix(.013,0,0,-0.013,78.72,0)\"><use xlink:href=\"#g113-48\"></use></g></svg><span></span><svg height=\"12.7178pt\" style=\"vertical-align:-3.42947pt\" version=\"1.1\" viewbox=\"84.03118380000001 -9.28833 20.326 12.7178\" width=\"20.326pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,84.081,0)\"><use xlink:href=\"#g113-79\"></use></g><g transform=\"matrix(.013,0,0,-0.013,94.991,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,99.476,0)\"><use xlink:href=\"#g113-42\"></use></g></svg></span> (where <svg height=\"11.439pt\" style=\"vertical-align:-2.15067pt\" version=\"1.1\" viewbox=\"-0.0498162 -9.28833 16.3975 11.439\" width=\"16.3975pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-92\"></use></g><g transform=\"matrix(.013,0,0,-0.013,4.485,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,11.752,0)\"><use xlink:href=\"#g113-94\"></use></g></svg> is the integral part of <span><svg height=\"6.1673pt\" style=\"vertical-align:-0.2063904pt\" version=\"1.1\" viewbox=\"-0.0498162 -5.96091 7.39387 6.1673\" width=\"7.39387pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-121\"></use></g></svg>,</span> <span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 22.173 10.2124\" width=\"22.173pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-79\"></use></g><g transform=\"matrix(.013,0,0,-0.013,14.542,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"25.7551838 -8.6359 9.204 10.2124\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.805,0)\"><use xlink:href=\"#g113-50\"></use></g><g transform=\"matrix(.013,0,0,-0.013,32.045,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"37.1381838 -8.6359 9.204 10.2124\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,37.188,0)\"><use xlink:href=\"#g113-51\"></use></g><g transform=\"matrix(.013,0,0,-0.013,43.428,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"48.521183799999996 -8.6359 9.204 10.2124\" width=\"9.204pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,48.571,0)\"><use xlink:href=\"#g113-52\"></use></g><g transform=\"matrix(.013,0,0,-0.013,54.811,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"59.9041838 -8.6359 9.205 10.2124\" width=\"9.205pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,59.954,0)\"></path></g><g transform=\"matrix(.013,0,0,-0.013,66.195,0)\"><use xlink:href=\"#g113-45\"></use></g></svg><span></span><span><svg height=\"10.2124pt\" style=\"vertical-align:-1.576501pt\" version=\"1.1\" viewbox=\"71.2881838 -8.6359 6.567 10.2124\" width=\"6.567pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,71.338,0)\"></path></g></svg>)</span></span> modulo <span><svg height=\"15.0208pt\" style=\"vertical-align:-3.429399pt\" version=\"1.1\" viewbox=\"-0.0498162 -11.5914 12.784 15.0208\" width=\"12.784pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-113\"></use></g><g transform=\"matrix(.0091,0,0,-0.0091,7.71,-5.741)\"><use xlink:href=\"#g50-51\"></use></g></svg>.</span> When <span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"-0.0498162 -8.6359 22.173 8.8423\" width=\"22.173pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,0,0)\"><use xlink:href=\"#g113-79\"></use></g><g transform=\"matrix(.013,0,0,-0.013,14.542,0)\"><use xlink:href=\"#g117-34\"></use></g></svg><span></span><span><svg height=\"8.8423pt\" style=\"vertical-align:-0.2064009pt\" version=\"1.1\" viewbox=\"25.7551838 -8.6359 6.429 8.8423\" width=\"6.429pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g transform=\"matrix(.013,0,0,-0.013,25.805,0)\"><use xlink:href=\"#g113-50\"></use></g></svg>,</span></span> the results we obtain are the results of Zhao and Xia and Cai.","PeriodicalId":54214,"journal":{"name":"Journal of Mathematics","volume":"37 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Congruences Involving Special Sums of Triple Reciprocals\",\"authors\":\"Zhongyan Shen\",\"doi\":\"10.1155/2024/8445635\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Define the sums of triple reciprocals <span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.68632 35.781 15.5493\\\" width=\\\"35.781pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,8.996,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,13.494,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,20.02,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,28.15,0)\\\"></path></g></svg><span></span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"39.363183799999995 -9.68632 54.435 15.5493\\\" width=\\\"54.435pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,39.413,.007)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,49.2,3.466)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,51.748,3.466)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,57.308,3.466)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,61.222,3.466)\\\"><use xlink:href=\\\"#g54-36\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,66.782,3.466)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,71.414,3.466)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,76.974,3.466)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,82.252,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,88.492,0)\\\"></path></g></svg><span></span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"93.8031838 -9.68632 17.77 15.5493\\\" width=\\\"17.77pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,93.853,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,96.358,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,102.119,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,108.659,0)\\\"></path></g></svg><span></span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"113.75218380000001 -9.68632 6.513 15.5493\\\" width=\\\"6.513pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,113.802,0)\\\"><use xlink:href=\\\"#g113-106\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,117.351,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"122.4441838 -9.68632 8.464 15.5493\\\" width=\\\"8.464pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,122.494,0)\\\"><use xlink:href=\\\"#g113-107\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,127.994,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"133.0871838 -9.68632 17.802 15.5493\\\" width=\\\"17.802pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,133.137,0)\\\"><use xlink:href=\\\"#g113-108\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,143.308,0)\\\"></path></g></svg><span></span><span><svg height=\\\"15.5493pt\\\" style=\\\"vertical-align:-5.86298pt\\\" version=\\\"1.1\\\" viewbox=\\\"154.5211838 -9.68632 6.697 15.5493\\\" width=\\\"6.697pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,154.571,0)\\\"><use xlink:href=\\\"#g113-50\\\"></use></g></svg>.</span></span> Zhao discovered the following curious congruence for any odd prime <span><svg height=\\\"10.2124pt\\\" style=\\\"vertical-align:-3.42943pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -6.78297 7.83752 10.2124\\\" width=\\\"7.83752pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"></path></g></svg>,</span> <span><svg height=\\\"14.7625pt\\\" style=\\\"vertical-align:-5.47417pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 36.965 14.7625\\\" width=\\\"36.965pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,8.996,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,13.494,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,21.204,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,29.334,0)\\\"></path></g></svg><span></span><svg height=\\\"14.7625pt\\\" style=\\\"vertical-align:-5.47417pt\\\" version=\\\"1.1\\\" viewbox=\\\"40.5471838 -9.28833 86.298 14.7625\\\" width=\\\"86.298pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,40.597,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,48.228,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,54.468,0)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,62.216,3.132)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,67.65,3.132)\\\"></path></g><g transform=\\\"matrix(.0091,0,0,-0.0091,73.21,3.132)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,78.197,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,82.695,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,93.289,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,100.037,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,111.346,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,119.056,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,123.554,0)\\\"></path></g></svg><span></span></span> Xia and Cai extended the above congruence to modulo <span><svg height=\\\"17.0656pt\\\" style=\\\"vertical-align:-5.474199pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -11.5914 15.621 17.0656\\\" width=\\\"15.621pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,7.71,-5.741)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,12.657,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span><svg height=\\\"17.0656pt\\\" style=\\\"vertical-align:-5.474199pt\\\" version=\\\"1.1\\\" viewbox=\\\"17.750183800000002 -11.5914 36.965 17.0656\\\" width=\\\"36.965pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,17.8,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,26.796,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,31.294,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,39.004,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,47.134,0)\\\"><use xlink:href=\\\"#g117-35\\\"></use></g></svg><span></span><svg height=\\\"17.0656pt\\\" style=\\\"vertical-align:-5.474199pt\\\" version=\\\"1.1\\\" viewbox=\\\"58.347183799999996 -11.5914 41.564 17.0656\\\" width=\\\"41.564pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,58.397,0)\\\"><use xlink:href=\\\"#g113-50\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,64.637,0)\\\"><use xlink:href=\\\"#g113-51\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,70.877,0)\\\"><use xlink:href=\\\"#g113-67\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,78.625,3.132)\\\"><use xlink:href=\\\"#g50-113\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,84.058,3.132)\\\"><use xlink:href=\\\"#g54-33\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,89.619,3.132)\\\"><use xlink:href=\\\"#g50-52\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,94.605,0)\\\"><use xlink:href=\\\"#g113-48\\\"></use></g></svg><span></span><svg height=\\\"17.0656pt\\\" style=\\\"vertical-align:-5.474199pt\\\" version=\\\"1.1\\\" viewbox=\\\"99.9161838 -11.5914 89.604 17.0656\\\" width=\\\"89.604pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,99.966,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,104.464,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,115.08,0)\\\"><use xlink:href=\\\"#g117-33\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,125.617,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,131.857,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,139.26,0)\\\"><use xlink:href=\\\"#g117-33\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,149.797,0)\\\"><use xlink:href=\\\"#g113-52\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,156.037,0)\\\"><use xlink:href=\\\"#g113-67\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,163.785,3.132)\\\"><use xlink:href=\\\"#g50-51\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,168.217,3.132)\\\"><use xlink:href=\\\"#g50-113\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,173.65,3.132)\\\"><use xlink:href=\\\"#g54-33\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,179.21,3.132)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,184.214,0)\\\"><use xlink:href=\\\"#g113-48\\\"></use></g></svg><span></span><svg height=\\\"17.0656pt\\\" style=\\\"vertical-align:-5.474199pt\\\" version=\\\"1.1\\\" viewbox=\\\"189.52518379999998 -11.5914 90.404 17.0656\\\" width=\\\"90.404pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,189.575,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,194.073,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,204.689,0)\\\"><use xlink:href=\\\"#g117-33\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,215.225,0)\\\"><use xlink:href=\\\"#g113-51\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,221.465,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,225.963,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,230.461,0)\\\"><use xlink:href=\\\"#g121-107\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,241.056,0)\\\"><use xlink:href=\\\"#g121-109\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,247.803,0)\\\"><use xlink:href=\\\"#g121-98\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,259.112,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.0091,0,0,-0.0091,266.822,-5.741)\\\"><use xlink:href=\\\"#g50-51\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,271.769,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,276.267,0)\\\"><use xlink:href=\\\"#g113-45\\\"></use></g></svg><span></span></span> where <span><svg height=\\\"11.5434pt\\\" style=\\\"vertical-align:-3.42945pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -8.11395 18.973 11.5434\\\" width=\\\"18.973pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,11.342,0)\\\"></path></g></svg><span></span><svg height=\\\"11.5434pt\\\" style=\\\"vertical-align:-3.42945pt\\\" version=\\\"1.1\\\" viewbox=\\\"22.555183800000002 -8.11395 6.419 11.5434\\\" width=\\\"6.419pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,22.605,0)\\\"></path></g></svg></span> is a prime. In this paper, we consider the congruences about <span><svg height=\\\"12.7178pt\\\" style=\\\"vertical-align:-3.42947pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 60.406 12.7178\\\" width=\\\"60.406pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-91\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,8.996,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,13.494,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,17.979,0)\\\"><use xlink:href=\\\"#g113-41\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,22.477,0)\\\"><use xlink:href=\\\"#g113-113\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,33.093,0)\\\"><use xlink:href=\\\"#g117-33\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,43.63,0)\\\"><use xlink:href=\\\"#g113-50\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,52.775,0)\\\"></path></g></svg><span></span><svg height=\\\"12.7178pt\\\" style=\\\"vertical-align:-3.42947pt\\\" version=\\\"1.1\\\" viewbox=\\\"63.2611838 -9.28833 20.765 12.7178\\\" width=\\\"20.765pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,63.311,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,74.222,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,78.72,0)\\\"><use xlink:href=\\\"#g113-48\\\"></use></g></svg><span></span><svg height=\\\"12.7178pt\\\" style=\\\"vertical-align:-3.42947pt\\\" version=\\\"1.1\\\" viewbox=\\\"84.03118380000001 -9.28833 20.326 12.7178\\\" width=\\\"20.326pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,84.081,0)\\\"><use xlink:href=\\\"#g113-79\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,94.991,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,99.476,0)\\\"><use xlink:href=\\\"#g113-42\\\"></use></g></svg></span> (where <svg height=\\\"11.439pt\\\" style=\\\"vertical-align:-2.15067pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -9.28833 16.3975 11.439\\\" width=\\\"16.3975pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g transform=\\\"matrix(.013,0,0,-0.013,0,0)\\\"><use xlink:href=\\\"#g113-92\\\"></use></g><g transform=\\\"matrix(.013,0,0,-0.013,4.485,0)\\\"></path></g><g transform=\\\"matrix(.013,0,0,-0.013,11.752,0)\\\"><use xlink:href=\\\"#g113-94\\\"></use></g></svg> is the integral part of <span><svg height=\\\"6.1673pt\\\" style=\\\"vertical-align:-0.2063904pt\\\" version=\\\"1.1\\\" viewbox=\\\"-0.0498162 -5.96091 7.39387 6.1673\\\" width=\\\"7.39387pt\\\" xmlns=\\\"http://www.w3.org/2000/svg\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g 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引用次数: 0
摘要
定义三重倒数之和.对于任意奇素数 , 夏和蔡发现了如下奇特的同余式 将上述同余式推广到模中 , 其中 , 是素数.在本文中，我们考虑的是关于 (其中是 , 的积分部分) modulo 的同余式.当 , 时，我们得到的结果是赵和夏和蔡的结果。本文章由计算机程序翻译，如有差异，请以英文原文为准。
Congruences Involving Special Sums of Triple Reciprocals
Define the sums of triple reciprocals . Zhao discovered the following curious congruence for any odd prime , Xia and Cai extended the above congruence to modulo where is a prime. In this paper, we consider the congruences about (where is the integral part of , ) modulo . When , the results we obtain are the results of Zhao and Xia and Cai.
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来源期刊
Journal of Mathematics
Mathematics-General Mathematics
CiteScore
2.50
自引率
14.30%
发文量
0
期刊介绍：
Journal of Mathematics is a broad scope journal that publishes original research articles as well as review articles on all aspects of both pure and applied mathematics. As well as original research, Journal of Mathematics also publishes focused review articles that assess the state of the art, and identify upcoming challenges and promising solutions for the community.
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