Tag Archives: mathematical entity

Two exceedingly simple sums related to triangular numbers

This note records some elementary arithmetic pertaining to triangular numbers for bālabodhana. In our youth we found that having a flexible attitude was good thing while obtaining closed forms for simple sums: for some sums geometry (using methods of proofs … Continue reading

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Johannes Germanus Regiomontanus and his rod

Even before we had become acquainted with the trigonometric sum and difference formulae or calculus are father had pointed to us that there was an optimal point at which one should stand to observe or photograph features on vertical structures, … Continue reading

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Modulo rugs of 3D functions

Consider a 3D function . Now evaluate it at each point of a integer lattice grid. Compute corresponding to each point and plot it as a color defined by some palette that suits your aesthetic. The consequence is a what … Continue reading

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A guilloche-like trigonometric tangle

Coprimality, i.e., the situation where the GCD of 2 integers is 1 is one of the fundamental expressions of complexity. In that situation, two numbers can never contain the other within themselves or in multiples of them by numbers smaller … Continue reading

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Some notes on the Henon-Heiles Hamiltonian system

Anyone familiar with dynamical systems knows of the Henon-Heiles (HH) system. What we are presenting here is well-known stuff about which reams of material have been written. However, we offer certain tricks for visualizing this system that make it easy … Continue reading

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Counting pyramids, squares and magic squares

Figure 1. Pyramidal numbers The following note provides some exceedingly elementary mathematics, primarily for bālabodhana. Sometime back we heard a talk by a famous contemporary mathematician (M. Bhargava) in which he described how as a kid he discovered for himself … Continue reading

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An arithmetic experiment and an unsolved problem

We realized that a simple arithmetic experiment we had performed in our youth is actually related to an unsolved problem in number theory. It goes thus: consider the sequence of natural numbers Then find the distance of to nearest prime … Continue reading

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Conic conquests: biographical and historical

PDF file of same article Studying mathematics with our father was not exactly an easy-going experience; nevertheless, it was the source of many a spark that inspired fruitful explorations and life-lessons. We recount one such thread here, and reflect on … Continue reading

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Generalizations of the prime sieve and Pi

PDF version for better reading Eratosthenes, the preeminent yavana philosopher of early Ptolemaic Egypt [footnote 1], composed a hymn to the god Hermes of which only some fragments have come down to us. This connection to Hermes is evidently related … Continue reading

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Two squares that sum to a cube

Introduction This note records an exploration that began in our youth with the simple arithmetic question: Sum of the squares of which pair integers yields a perfect cube? Some obvious cases immediately come to mind: . In both these cases … Continue reading

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Some Nārāyaṇa-like convergents and their geometric and trigonometric connections

While playing with an iterative geometric construction in our youth we discovered for ourselves a particular right triangle whose sides are in the proportion , where is the Golden Ratio. This triangle is of course famous as being the basis … Continue reading

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A modern glance at Nārāyaṇa-paṇḍita’s combinatorics-1

For improved reading experience one may use the PDF version. Students of the history of Hindu mathematics are well-acquainted with Nārāyaṇa-paṇḍita’s sophisticated treatment of various aspects of combinatorics and integer sequences in his Gaṇita-kaumudī composed in 1356 CE. In that … Continue reading

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Pearl necklaces for Maheśvara

Śrīpati’s pearl necklace for Maheśvara The brāhmaṇa Śrīpati of the Kāśyapa clan was a soothsayer from Rohiṇīkhaṇḍa, which is in the modern Buldhana district of Maharashtra state. Somewhere between 1030 to 1050 CE he composed several works on mathematics, astronomy … Continue reading

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Discovering bronze in the characteristic ellipse of right triangles

The arithmetic mean square of a right triangle An entire family of right triangles that includes all the different forms of right triangles defined in terms of the proportion of their legs can be obtained by setting their altitude to … Continue reading

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An apparition of Mordell

Consider the equation: where is a positive integer 1, 2, 3… For a given , will the above equation have integer solutions and, if yes, what are they and how many? We have heard of accounts of people receiving solutions … Continue reading

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Visualizing the Hindu divisibility test

Prologue This article continues on the themes covered by the last two (here and here) relating to factorization and the primitive root modulo of a prime number. Early in ones education one learns the divisibility tests for the first few … Continue reading

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Division-multiplication parabolas, triplications, and quadratic residues

Introduction Many strands of our investigations on conic-generating integer sequences, word fractals and cellular automaton models for pattern formation came together in an unexpected manner while investigating a simple integer sequence. While some of these connections have have been known … Continue reading

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Residues of squares, sequence curiosities and parabolas galore

Squares and their residues This is an exploration of number triangles in the same vein as some other such we have previously described . It resulted in some observations that seemed interesting to us. Some are perhaps trivial but some … Continue reading

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A note on the least prime divisor sequences of 2p plus or minus 1

Let be the sequence of prime numbers: 2, 3, 5, 7… Define the sequences such that . Then sequence is defined such that is the lowest prime divisor (LPD) of and sequence is defined so that is the LPD of … Continue reading

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Convergence to a palindrome

This is a brief account of a sequence we constructed inspired by Dattatreya Ramachandra Kaprekar. It is not known to us if he had discovered it in one of his obscure publications from a small town in the Maharatta country. … Continue reading

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The remarkable behavior of a map displaying derived from a simple model for a biological conflict

One of the simplest yet profound mathematical models for biological growth emerged sometime in the middle of the 1800s due to the work of Verhulst. It describes population growth thus: let be the population of the organism at time . … Continue reading

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The incredible beauty of certain Hamiltonian mappings

In our teens we studied Hamiltonian functions a little bit as part of our attempt to understand classical and quantum physics. A byproduct of it was a superficial interest in the geometry of some of the mappings arising from such … Continue reading

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The square root spiral and the Gamma function: entwined analogies

The topic discussed here is something on which considerable serious mathematical literature has published by P.J Davis, W. Gautschi and others. This partly historical narration is just a personal account of our journey through the same as a non-mathematician. As … Continue reading

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Cobwebs on the golden hyperbola and parabola

The material presented here is rather trivial to those who have spent even a small time looking at chaotic systems. Nevertheless, we found it instructive when we first discovered it for ourselves while studying conics. Hence, as part of recording … Continue reading

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The magic of the deva-ogdoad

Classical Hindu tradition holds that the ogdoad of deva-s corresponding to their directions is: Indra: East; Agni: Southeast; Yama: South; Nirṛti: Southwest; Varuṇa: West; Vāyu: Northwest; Kubera: North; Īśāna: Northeast. The central position might be occupied in certain traditions by … Continue reading

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Trigonometric tangles-3: the fractals

See also: https://manasataramgini.wordpress.com/2016/05/06/the-astroid-the-deltoid-and-the-fish-within-the-fish/ This exploration began in days of youth shortly after we learned about complex numbers. It culminated only much later in adulthood when we discovered for ourselves a class of fractal curves related to a celebrated curve discovered … Continue reading

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Trigonometric tangles

Let us define a define the trigonometric tangle as the following parametric function: where can be a rational number or an irrational number. and are any real number. If is a rational number and then we get a tangle petals … Continue reading

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Syllable, number and rules in the ideal realm

This note is neither meant to be complete exposition of this matter nor a complete view of all what we have realized in this regard. Nor can it be completely understood by those who are not insiders of the tradition. … Continue reading

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A golden construction

Anyone with even a small fancy for geometrical matters would have at some point in their lives played with the golden ratio (). Indeed, we too have had our share of fun and games with the golden ratio. In course … Continue reading

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Computational driftwood

The above object is a gnarled piece of driftwood. Only that it never floated in the sea and exists only in the virtual realm – a pure mathematical abstraction – a combination of three functions. Some would say it is … Continue reading

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