[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-344502-105":59,"doc-detail-344502-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","the-3n1-conjecture-behaviour-of-the-stopping-time-function-extended-essay","The 3n+1 Conjecture - Behaviour of the Stopping Time Function - Extended Essay","","The Collatz (3n+1) conjecture asserts that iterating the function on any starting positive integer eventually reaches the cycle 2-1, with all trajectories falling into the repeating 2,1 pattern. The essay studies this process by introducing the total stopping time function, σ(n), defined as the number of iterations until first reaching 1. It outlines trajectories via altitudes, summarizes established background and motivation, and develops computational and explanatory approaches for observed stopping-time patterns, including congruences modulo 20(5)-1.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/the-3n1-conjecture-behaviour-of-the-stopping-time-function-extended-essay/344502/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/the-3n1-conjecture-behaviour-of-the-stopping-time-function-extended-essay/344502.png","ImageObject",300,407,{"name":92,"@type":93},"Valentina","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the 3n+1 (Collatz) conjecture claim about iterating T(n)?","Question",{"text":112,"@type":113},"It claims that starting from any positive integer, repeated application of the Collatz function eventually reaches the trivial cycle 2-1. After finitely many steps, the process falls into alternating between 2 and 1.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the total stopping time function σ(n) in this essay?",{"text":117,"@type":113},"σ(n) is defined as the number of altitudes (or iterations) in the forward orbit of a starting value until it first reaches altitude 1. It is used to quantify how long a number takes to enter the 2-1 cycle.",{"name":119,"@type":110,"acceptedAnswer":120},"What computational or analytical goal does the essay pursue?",{"text":121,"@type":113},"The essay aims to compute total stopping times for a wide range of starting values and identify patterns in σ(n). 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 \nThe Collatz conjecture answers this question affirmatively.It states that:  \n\"The process of iterating the Collatz function yields orbits which eventuallyreach the cycle 2-1,regardless of the positive integer initially chosen.\"  \nThe 3n+1Conjecture:Behaviour of the Stopping Time Function  \nAlex Lupsasca  \nMore formally,this is expressed as:  \nVn∈N>03i∈N:(ao=n→a₁=1)  \nIf the conjecture is false,it can only be because there is some natural numberwhich gives rise to a sequence that does not comprise the altitude 1.Such anorbit might enter a repeating cycle that excludes 1 or increase without bound.  \nIf the conjecture is true,an interesting aspect of the problem that lends itself toinvestigation is the \"total stopping time function,\"which I shall introduce andformally develop over the course of this essay in ","cbCaifkeUPkT9VUg","https://ap.wps.com/l/cbCaifkeUPkT9VUg","pdf",8120974,29,"English","# Abstract\n# Statement of the Problem\n## Total stopping time function and motivation\n## Collatz function definition and trajectories\n## Trigonometric approach and function extension\n## Simplification for positive integers","[{\"question\":\"What does the 3n+1 (Collatz) conjecture claim about iterating T(n)?\",\"answer\":\"It claims that starting from any positive integer, repeated application of the Collatz function eventually reaches the trivial cycle 2-1. After finitely many steps, the process falls into alternating between 2 and 1.\"},{\"question\":\"What is the total stopping time function σ(n) in this essay?\",\"answer\":\"σ(n) is defined as the number of altitudes (or iterations) in the forward orbit of a starting value until it first reaches altitude 1. It is used to quantify how long a number takes to enter the 2-1 cycle.\"},{\"question\":\"What computational or analytical goal does the essay pursue?\",\"answer\":\"The essay aims to compute total stopping times for a wide range of starting values and identify patterns in σ(n). It then seeks an explanation for observed behavior, based on congruences mentioned in the work.\"}]","The 3n+1 Conjecture - Behaviour of the Stopping Time Function - Extended Essay | PDF",1790054119,73]