[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83722-en":3,"doc-seo-83722-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83722,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Calculating the Floor of y 1/m","Two integer algorithms are presented to compute ⌊y1/m⌋ using Newton–Raphson ideas for natural numbers y>2 and m>1. The goal is to identify perfect powers y=xm and recover the base x by computing Y=⌊y1/m⌋ and verifying whether Ym=y. The work contrasts with binary-search exponent checks used in number-theory algorithms such as Miller’s and AKS, emphasizing integer-based iterative refinement and improved practicality for bit-model arithmetic.","arXiv :2607 .03462v1 [math .NT] 3 Jul 2026  \nCalculating the floor of y 1/m  \nAlexandros V. Gerbessiotis∗  \nJuly 7, 2026  \nAbstract  \nWe present two algorithms based on the Newton-Raphson method to calculate ⌊y1/m⌋ for natural integer numbers y > 2 and m > 1. One could use such an algorithm to establish whether y is an integer power of an integer in number theory problems, even though binary search methods are traditionally considered simpler to implement.  \n1 Introduction  \nIn several number theory algorithms we would like to determine whether natural number y is a perfect power that is, there exist two natural numbers x and m > 1 such that y = xm. The primality testing algorithms of Miller [3] [4] and AKS primality testing [1] utilize such an identification method in the introductory steps of those algorithms. The traditional approach to find x, m is to use binary search. For y = xm and m > 1 implies y = xm ≥ 2m and therefore m ≤ lg y. Therefore one tests for each integer m between integer two and the integer closer to lg y (or the floor of it) whether there exists an integer x such that y = xm by performing a binary search of x on the integer interval between two and y. The exponentiation algorithm with repeated doubling can calculate xm efficiently within the efficiency requirements of those two algorithms. Furthermore, one need only search for prime numbers m in the given range and by doing so one can further optimize the running time of the approach. In general the running time for perfect power identification contributes low order terms to the overall running time either of Miller’s primality testing algorithms, or the AKS algorithm. The binary search-based method is not computationally efficient in the bit model of computation, where bit-based arithmetic computations are used to express the arithmetic complexity of an algorithm. Yet it is quite practical.  \nIt is known that other methods can be used to solve the perfect power identification problem as implied in [2](page 59) . For example, Newton-Raphson based methods have been used to determine the m-th root y of an arbitrary real number y. We utilize such a method to calculate ⌊y1/m ⌋ . Infact we present two different algorithms: they differ only on the way they derive an integer-based version of Newton-Raphson. Having computed Y = ⌊y1/m ⌋ one needs only test whether Y m = y to determine not only whether y is a perfect power but calculate x = Y.  \nSuch techniques using Newton-Raphson for integers problems are not new e.g. see [2] . Our intent is just to provide an archival reference to such methods rather than making claims on their novelty.  \n∗ CS Department, New Jersey Institute of Technology, Newark, NJ 07102, USA. Email: [alexg@njit.edu](alexg@njit.edu)  \n2 First approach  \nThe first approach is a quite straightforward adaptation of a real number Newton-Raphson method  1   \nfor calculating y m . It is modified into an integer-based Newton-Raphson method for calculating jy ~~1~~mk. We call this approach Algorithm 1 . It is subsequently modified and refined into what we call Algorithm 2 .  \nLemma 1 . Given a natural integer number y > 2 and a natural integer number m > 1, Algorithm 1 determines a natural integer number x such that  \nx = jy ~~1~~mk .  \nProof.  \nIn the remainder, lg y denotes the logarithm of y base two. We are going to use the following inequalities related to the floor and ceiling functions.  \nx − 1 \u003C ⌊x⌋ ≤ x ≤ ⌈x⌉ \u003C x + 1 . (1)  \n1 + ⌊x⌋ ≥ ⌈x⌉ . (2)  \nThe symbol ⌊ \u003C 0⌋ will denote a negative quantity. The symbol ⌈> 0⌉ will denote a positive quantity. In either case we are only interested in the sign, but not the quantity itself. The Newton-Raphson method is highlighted below. For more information see for example [5] .  \n(a) Newton-Raphson iterative method: an overview.  \nGiven f (x) in Eq.(3) that follow,  \nf (x) = y − xm , (3)  \nits solution for f (x) = 0 is x = y 1/m. Let Y = y 1/m. Moreover we derive that f′(x) = −mxm−1 . Let us then use the Ne","cbCaicROY67uyVde","https://ap.wps.com/l/cbCaicROY67uyVde","pdf",319844,4,1,14,"English","en",105,"# Abstract\n# Introduction\n# First approach\n## Lemma 1\n## Newton-Raphson iterative method\n## Integer Newton-Raphson\n## Choice of the initial value x0\n## Iteration i=0 to i+1=1","[{\"question\":\"What problem do the two proposed algorithms solve?\",\"answer\":\"They compute the integer value ⌊y1/m⌋ for natural numbers y\\u003e2 and m\\u003e1, enabling perfect power identification y=xm.\"},{\"question\":\"How does computing ⌊y1/m⌋ help determine whether y is a perfect power?\",\"answer\":\"After computing Y=⌊y1/m⌋, it suffices to test whether Ym=y; if true, then y is a perfect power and x=Y.\"},{\"question\":\"How do the algorithms relate to the Newton–Raphson method?\",\"answer\":\"They adapt Newton–Raphson to an integer setting, using an integer-based recurrence derived from the real-variable iteration and designed to keep intermediate values integral.\"}]",1784189979,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"calculating-the-floor-of-y-1m","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/calculating-the-floor-of-y-1m/83722/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem do the two proposed algorithms solve?","Question",{"text":75,"@type":76},"They compute the integer value ⌊y1/m⌋ for natural numbers y>2 and m>1, enabling perfect power identification y=xm.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does computing ⌊y1/m⌋ help determine whether y is a perfect power?",{"text":80,"@type":76},"After computing Y=⌊y1/m⌋, it suffices to test whether Ym=y; if true, then y is a perfect power and x=Y.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the algorithms relate to the Newton–Raphson method?",{"text":84,"@type":76},"They adapt Newton–Raphson to an integer setting, using an integer-based recurrence derived from the real-variable iteration and designed to keep intermediate values integral.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]