{{Short description|Indian mathematician (about 1325–1400)}} {{About|Indian mathematician Narayana Pandita|Narayana Pandita, the writer of the book Hitopadesh|Narayan Pandit}} {{Use dmy dates|date=October 2024}} '''Nārāyaṇa Paṇḍita''' ({{langx|sa|नारायण पण्डित}}) (1340–1400{{Cite web |title=Narayana - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Narayana/ |access-date=2022-10-03 |website=Maths History}}) was an Indian [[mathematician]]. [[Kim Plofker|Plofker]] writes that his texts were the most significant Sanskrit mathematics treatises after those of [[Bhaskara II]], other than the [[Kerala school of astronomy and mathematics|Kerala school]].{{rp|52}} He wrote the ''[[Ganita Kaumudi]]'' (lit. "Moonlight of mathematics") in 1356 about mathematical operations. The work anticipated many developments in [[combinatorics]]. ==Life and Works== About his life, the most that is known is that: {{blockquote|His father’s name was Nṛsiṃha or Narasiṃha, and the distribution of the manuscripts of his works suggests that he may have lived and worked in the northern half of India.}} Narayana Pandit wrote two works, an arithmetical treatise called ''Ganita Kaumudi'' and an [[algebra]]ic treatise called ''Bijaganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of [[Bhaskara II]]'s [[Lilavati]], titled ''Karmapradipika'' (or ''Karma-Paddhati''). Although the ''Karmapradipika'' contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and [[magic square]]s. Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order [[indeterminate equation]] ''nq''2 + 1 = ''p''2 ([[Pell's equation]]), solutions of indeterminate [[Degree of a polynomial|higher-order equations]], mathematical operations with [[0 (number)|zero]], several [[geometry|geometrical]] rules, methods of integer factorization, and a discussion of magic squares and similar figures. Narayana has also made contributions to the topic of [[cyclic quadrilateral]]s. Narayana is also credited with developing a method for [[Permutation#Generation in lexicographic order|systematic generation of all permutations]] of a given sequence. ==Narayana's cows sequence== In his ''Ganita Kaumudi'' Narayana proposed the following problem on a herd of cows and calves: {{blockquote|A cow produces one calf every year. Beginning in its fourth year, each calf produces one calf at the beginning of each year. How many cows and calves are there altogether after 20 years?}} Translated into the modern mathematical language of [[Recurrence relation|recurrence sequences]]: :{{math|1= N''n'' = N''n''-1 + N''n''-3}} for {{math|''n'' > 2}}, with initial values :{{math|1= N0 = N1 = N2 = 1}}. The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... {{OEIS|A000930}}. The limit ratio between consecutive terms is the [[supergolden ratio]]. The recurrence on {{math|N''n''-1 + N''n''-''k''}} puts Narayana's cows and the supergolden ratio as the next in a series of sequences starting with ''k'' = 1 the [[powers of two]] with 2, and ''k'' = 2 the [[Fibonacci sequence]] with the [[golden ratio]], which are used in computing to make [[Buddy memory allocation|buddy allocators]]. ==See also== *[[Fibonacci sequence]] *[[Golden ratio]] *[[Supergolden ratio]] *[[Archimedes cattle problem]] *[[Pell's equation]] ==References== {{Reflist|refs= {{citation | author=[[Kim Plofker]] | title=Mathematics in India: 500 BCE–1800 CE | title-link= Mathematics in India (book) | year=2009 | place = Princeton, NJ | publisher=Princeton University Press | isbn= 978-0-691-12067-6}} J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Narayana.html Narayana] {{webarchive|url=https://web.archive.org/web/20080124154540/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Narayana.html |date=2008-01-24 }}, ''[[MacTutor History of Mathematics archive]]''.{{Unreliable source?|date=March 2011}} Ian G. Pearce (2002). [http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_2.html Mathematicians of Kerala] {{webarchive|url=https://web.archive.org/web/20081219151102/http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_2.html |date=2008-12-19 }}. ''MacTutor History of Mathematics archive''. [[University of St Andrews]].{{Unreliable source?|date=March 2011}} {{citation | last=Kusuba|first=Takanori | contribution=Indian Rules for the Decomposition of Fractions | year=2004 | title=Studies in the History of the Exact Sciences in Honour of [[David Pingree]] | publisher=[[Brill Publishers|Brill]] | isbn=9004132023 | issn=0169-8729 | editor1=Charles Burnett | editor2=Jan P. Hogendijk | editor3=Kim Plofker |display-editors = 3 | editor4=Michio Yano | page = 497}} {{cite journal | last1 = Hirschberg | first1 = Daniel S. | author-link1 = Dan Hirschberg | title = A class of dynamic memory allocation algorithms | journal = Commun. ACM | volume = 16 | issue = 10 | pages = 615–618 | publisher = Association for Computing Machinery | location = New York | year = 1973 | url = https://doi.org/10.1145/362375.362392 | issn = 0001-0782 | doi = 10.1145/362375.362392}} {{cite web | title = Generalized Fibonacci Memory Allocator | author = naens | date = April 30, 2019 | url = https://dev.to/naens/generalized-fibonacci-memory-allocator-2fja | publisher = Dev.to | access-date = August 2, 2025}} }} {{Indian mathematics}} {{authority control}} {{DEFAULTSORT:Pandit, Narayana}} [[Category:1340 births]] [[Category:1400 deaths]] [[Category:Indian Hindus]] [[Category:14th-century Indian mathematicians]] {{India-scientist-stub}} {{asia-mathematician-stub}}