{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-06-05T00:10:21.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-06-05T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":1215,"title":"Diophantine Equations (Inspired by Project Euler, problem 66)","description":"Consider the quadratic Diophantine equation of the form:\r\n\r\nx^2 – Dy^2 = 1\r\n\r\nWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.  It can be assumed that there are no solutions in positive integers when D is square.\r\n\r\nGiven a value of D, find the minimum value of X that gives a solution to the equation.","description_html":"\u003cp\u003eConsider the quadratic Diophantine equation of the form:\u003c/p\u003e\u003cp\u003ex^2 – Dy^2 = 1\u003c/p\u003e\u003cp\u003eWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.  It can be assumed that there are no solutions in positive integers when D is square.\u003c/p\u003e\u003cp\u003eGiven a value of D, find the minimum value of X that gives a solution to the equation.\u003c/p\u003e","function_template":"function y = Diophantine(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2; y_correct = 3; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 151; y_correct = 1728148040; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 61; y_correct = 1766319049; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 66; y_correct = 65; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 12000; y_correct = 13007560326001; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 2345; y_correct = 15129001; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nj=[10:10:90 110:10:200]; v=arrayfun(@(x) Diophantine(j(x)),1:length(j));\r\ny_correct=[19 9 11 19 99 31 251 9 19 21 11 6499 71 49 721 339 161 52021 99];\r\nassert(all(isequal(v,y_correct)))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":8,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":70,"test_suite_updated_at":"2013-10-17T19:30:05.000Z","rescore_all_solutions":false,"group_id":25,"created_at":"2013-01-18T19:13:15.000Z","updated_at":"2026-05-06T03:31:21.000Z","published_at":"2013-01-18T19:13:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the quadratic Diophantine equation of the form:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2 – Dy^2 = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1. It can be assumed that there are no solutions in positive integers when D is square.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a value of D, find the minimum value of X that gives a solution to the equation.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"problems":[{"id":1215,"title":"Diophantine Equations (Inspired by Project Euler, problem 66)","description":"Consider the quadratic Diophantine equation of the form:\r\n\r\nx^2 – Dy^2 = 1\r\n\r\nWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.  It can be assumed that there are no solutions in positive integers when D is square.\r\n\r\nGiven a value of D, find the minimum value of X that gives a solution to the equation.","description_html":"\u003cp\u003eConsider the quadratic Diophantine equation of the form:\u003c/p\u003e\u003cp\u003ex^2 – Dy^2 = 1\u003c/p\u003e\u003cp\u003eWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1.  It can be assumed that there are no solutions in positive integers when D is square.\u003c/p\u003e\u003cp\u003eGiven a value of D, find the minimum value of X that gives a solution to the equation.\u003c/p\u003e","function_template":"function y = Diophantine(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 2; y_correct = 3; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 151; y_correct = 1728148040; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 61; y_correct = 1766319049; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 66; y_correct = 65; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 12000; y_correct = 13007560326001; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nx = 2345; y_correct = 15129001; assert(isequal(Diophantine(x),y_correct))\r\n%%\r\nj=[10:10:90 110:10:200]; v=arrayfun(@(x) Diophantine(j(x)),1:length(j));\r\ny_correct=[19 9 11 19 99 31 251 9 19 21 11 6499 71 49 721 339 161 52021 99];\r\nassert(all(isequal(v,y_correct)))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":8,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":70,"test_suite_updated_at":"2013-10-17T19:30:05.000Z","rescore_all_solutions":false,"group_id":25,"created_at":"2013-01-18T19:13:15.000Z","updated_at":"2026-05-06T03:31:21.000Z","published_at":"2013-01-18T19:13:15.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the quadratic Diophantine equation of the form:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^2 – Dy^2 = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhen D=13, the minimal solution in x is 649^2 – 13×180^2 = 1. It can be assumed that there are no solutions in positive integers when D is square.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a value of D, find the minimum value of X that gives a solution to the equation.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"errors":[],"facets":[[{"value":"Functions I","count":1,"selected":false},{"value":"Project Euler III","count":1,"selected":false}],[{"value":"hard","count":1,"selected":false}]],"term":"tag:\"pell equation\"","page":1,"per_page":50,"sort":"map(difficulty_value,0,0,999) asc"}}