Nārāyaṇa Paṇḍita (Sanskrit:नारायण पण्डित) (1340–1400[1]) was an Indianmathematician.Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those ofBhaskara II, other than theKerala school.[2]: 52 He wrote theGanita Kaumudi (lit. "Moonlight of mathematics"[3]) in 1356[3] about mathematical operations. The work anticipated many developments incombinatorics.
About his life, the most that is known is that:[2]
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 calledGanita Kaumudi and analgebraic treatise calledBijaganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary ofBhaskara II'sLilavati, titledKarmapradipika (orKarma-Paddhati).[4] Although theKarmapradipika 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 andmagic squares.[4]
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 orderindeterminate equationnq2 + 1 =p2 (Pell's equation), solutions of indeterminatehigher-order equations, mathematical operations withzero, severalgeometrical rules, methods of integer factorization, and a discussion of magic squares and similar figures.[4] Narayana has also made contributions to the topic ofcyclic quadrilaterals.[5]Narayana is also credited with developing a method forsystematic generation of all permutations of a given sequence.
In hisGanita Kaumudi Narayana proposed the following problem on a herd of cows and calves:
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 ofrecurrence sequences:
with initial values
The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... (sequenceA000930 in theOEIS).The limit ratio between consecutive terms is thesupergolden ratio.
The recurrence onNn-1 + Nn-k puts Narayana's cows and the supergolden ratio as the next in a series of sequences starting withk = 1 thepowers of two with 2, andk = 2 theFibonacci sequence with thegolden ratio, which are used in computing to make buddy allocators.[6][7]
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