# Tag Archives: numbers

## A sampler of Ramanujan’s elementary results and their manifold ramifications

As we have remarked before, Ramanujan seemed as if channeling the world-conquering strides of Viṣṇu, when he single-handedly bridged the lacuna in Hindu mathematics from the days of the brāhmaṇa-s of the Cerapada to the modern era. Starting around the … Continue reading

## A catalog of attractors, repellors, cycles, and other oscillations of some common functional iterates

One of the reasons we became interested in functional iterates was from seeking an analogy for the effect of selective pressure on the mean values of a measurable biological trait in a population. Let us consider a biological trait under … Continue reading

## Some observations on the Lekkerkerker-Zeckendorf decomposition of integers

In our youth, we learned of a nice arithmetic theorem of Lekkerkerker (more popularly known after Zeckendorf; hereinafter L-Z) that relates to the famous Mātrā-meru sequence : 0, 1, 1, 2, 3, 5, 8… defined by the recurrence relationship . … Continue reading

## Some biographical reflections on visualizing irrationals

In our childhood, our father informed us that, though the school told us that , it was not valid. However, he added that for “small fractions” [Footnote 1] it was a great approximation. Moreover, the numerical problems, which we would … Continue reading

## Bhāskara’s dual square indeterminate equations

PDF for convenient reading Figure 1. Sum and difference of squares amounting to near squares. In course of our exploration of the bhūjā-koṭi-karṇa-nyāya in our early youth we had observed that there are examples of “near misses”: . Hence, we … Continue reading

## 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

## 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

## 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

## Chaotic behavior of some floor-squared maps

Consider the one dimensional maps of the form: , where is the fractional part of What will be evolution of a under this map when or ? We can see that for it will tend converge. However, the behavior is … Continue reading

## 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

## Difference of consecutive cubes, conics and a Japanese temple tablet

Introduction In our part of the world, someone with even a nominal knowledge of mathematics might be aware of the taxicab number made famous by the conversation of Ramanujan and Hardy: the smallest number that can be expressed as the … Continue reading

## Sequences related to maps based on simple fractional functions

One of the pleasures of an unstructured youth in the pre-computer era was what we called calculator games. As our father took his prized calculator with him to work we only got a little time with it in the evenings. … Continue reading

## Some notes on rational sector triangle triples

Rational points on a unit circle There are some events that happen in the course of ones life that might be considered historical or world-changing. One such event from our lifetime is the proving of the Last Theorem of Fermat … Continue reading

## 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

## 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

## A layman’s overview of the arithmetic of encryption

Life as an encryption-decryption cycle Encryption is a concept as old as life itself. The sequence of proteins, the primary purveyors of function in life as we know it, is encrypted within nucleic acids. It is decrypted by this remarkable … Continue reading

## 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

## The mean hyperbola and other mean functions

Let be two numbers such that, We use to construct a specific rectangular hyperbola using one of the following methods: Method-I (Figure 1: this is based on an approach we described earlier) Figure 1 1) Mark point , which will … Continue reading

## The geometric principles behind discrete dynamical systems based on the generalized Witch of Agnesi

Consider the family of curves defined by the equation following parametric equation , where and It defines a family of probability distribution functions (PDFs): This can be seen from the above equations because Figure 1 Examples of these PDFs are … Continue reading

## Reflections on our journey through the aliquot sums and sequences

The numerology of aliquot sums and perfect numbers The numerology of the Pythagorean sages among the old yavana-s is one of the foundations of science and mathematics as we know it. One remarkable class of numbers which they discovered were … Continue reading

## 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