Taneja's Number Theory Papers

Inder J. Taneja
Federal University of Santa Catarina
Ph.D. from Delhi University, India

In my previous post titled Fibonacci Sequence and Selfie Numbers, I referenced a paper by Inder J. Taneja with the same title. I mentioned too that he has published many interesting Number Theory related papers and in this post I aim to summarise some of them and provide links to them. Looking back at my previous posts I discovered that I had made reference to two of Taneja's papers in a post titled Selfie Numbers (March 2020). Let's begin.

• Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9

Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation and division. In both the situations there are no missing numbers, except one (10958) in the increasing case.

• Selfie Numbers: Consecutive Representations in Increasing and Decreasing Orders

In this work, the numbers have been written in terms of increasing and decreasing orders of the digits in a consecutive way. To write these numbers, the operations used are: addition, subtraction, multiplication, potentiation, division, factorial and square-root. We named these numbers as selfie numbers, because of the fact that they have same digits on both sides of the expressions.

• Single Digit Representations of Natural Numbers 

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation and division.

• Running Expressions in Increasing and Decreasing Orders of Natural Numbers Separated by Equality Signs 

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation, division.

• Different Types of Pretty Wild Narcissistic Numbers: Selfie Representations 

In this work, the numbers have been written in order of digits and their reverse, generally famous as ”pretty wild narcissistic numbers”. To write these numbers, the operations used are: addition, subtraction, multiplication, potentiation, division, factorial, square-root. For simplicity, these representations are named as selfie numbers. These representations have same digits on both sides of the expressions with the properties that, they are either in order of digits or in reverse order. The work is separated in different types, such as, Palindromic, Symmetrical consecutive, Sequential selfies, etc.

• Single Letter Representations of Natural Numbers, Palindromic Symmetries and Number Patterns 

This is first work of its kind. It brings representations of natural numbers from 0 to 3000 in terms of single letter a. For any value of letter a from 1 to 9, the result is always same. Four basic operations, i.e., addition, subtraction, multiplication and division are used to bring these representations. A separate section is dedicated to numbers with potentiation. Palindromic symmetries and number patterns in terms of letter a are also studied

• Representations of Palindromic, Prime, and Fibonacci Sequence Patterns 

This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime number patterns are also considered. Different classifications of palindromic patterns are considered, such as, palindromic decompositions, double symmetric patterns, number pattern decompositions, etc. Numbers patterns with power are also studied. Study towards Fibonacci sequence and its extensions is also made.

• Representations of Palindromic, Prime and Number Patterns 

This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime number patterns are also considered. Different classifications of palindromic patterns are considered, such as, palindromic decompositions, double symmetric patterns, number pattern decompositions, etc. Numbers patterns with power are also studied.

• Unified Selfie Numbers 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. In this paper, we worked with Selfie numbers having all the four ways of representations at the same time. These numbers are called ”unified Selfie numbers”.

• Patterns in Selfie Numbers 

The idea of this work is to bring patterns in Selfie numbers. This we have done in two different ways. One is in order of digits and second is in decreasing order. The is limited only up to six  digits. Up to five digits, we worked with square-root and factorial. For six digits the work is only for square-root.

• Selfie Numbers - I: Symmetrical and Unified Representations 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. This work is improvement over the above works specially in case of increasing and decreasing order of digits. Symmetrical consecutive and unified Selfie numbers are also presented.

• Selfie Numbers - II: Six Digits Symmetrical, Unified and Patterned Representations Without Factorial. 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. In this work we have obtained Selfie numbers having six digits with repetitions without use of factorial. Symmetrical consecutive and unified Selfie numbers are also presented.

• Selfie Numbers - III: With Factorial and Without Square-Root - Up To Five Digits 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. This work is restricted up to five digits only with factorial and without use of square-root. Studies including square-root can be seen in author’s work.

• Selfie Power Representations

This paper works with representations of numbers with same digits on both sides of the expressions. The representations are made with the power of same digits as of numbers using only addition and subtraction signs. This is done only for eight and nine different digits. 

• Crazy Power Representations of Natural Numbers

This paper works with representations of natural numbers from 0 to 11111 written in terms of expressions with additions, subtractions and exponents. Digits used are from 1 to 9 in such a way that for each number, there are same digits in bases and exponents with different permutations. Some numbers can be written in more than one way, but we have chosen with less possible expressions. 

• Flexible Power Narcissistic Numbers with Division

This paper works with extensions of narcissistic numbers in different situations. Extensions are made for positive and negative coefficients, fixed and flexible powers. The idea is extended for narcissistic numbers with division. Here also different situations are considered, such as, positive and negative coefficients, fixed and flexible powers. Comparison with previous known numbers are also given. 

• Floor Function and Narcissistic Numbers with Division 

Narcissistic numbers are famous in literature. There are very few narcissistic numbers with division. In this work we brought some narcissistic number with division in terms of floor function.

• Double Sequential Representations of Natural Numbers - I

This work brings representations of natural numbers in two different ways. In both the representations same digits are used always ending in 0 such as, 210, 3210, etc.. 

• Flexible Power Selfie Numbers - I

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked up to width 7, where up to width 6 there are repetition in digits. From width 7 onwards, results are without any repetition. 8 and 9 width numbers are done in subsequent papers.

• Flexible Power Selfie Numbers - II

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked with width 8 numbers.

• Flexible Power Selfie Numbers - III

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked with width 9 numbers. 

• Double Sequential Representations of Natural Numbers - II

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations, the same digits from 7 to 0 are used in decreasing order. 

• Pyramidical Representations of Natural Numbers

This work brings representations of natural numbers in two different ways. In both the representations same digits are used always ending in 0 such as, 210, 3210, etc.. 

• Selfie Fractions: Addable

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. This work brings addable fractions in different situations. One for multiple choices, and second for single representations. In each fraction, the numerator less than denominator, and there is no repetition of digits. 

• Selfie Fractions: Dottable and Potentiable

A dottable fraction is a proper fraction where multiplication signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same happens with potentiation. In this case we call it potentiable fraction. This work brings dottable fractions and dottable fractions with potentiation in different situations without repetition of digits. The work is limited up to six digits in the denominator. 

• Selfie Fractions: Addable and Dottable Together

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for dottable fractions, i.e., instead of additions we have multiplication. In this work we have written fractions having both the operations, i.e., addition and multiplication. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator is less than denominator. 

• Equivalent Selfie Fractions: Dottable, Addable and Subtractable

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for subtractable fractions, i.e., instead of additions we have substraction. In this work we have written symmetric equivalent fractions having both the operations, i.e., one side is addition and another side is subtraction written in symmetric way. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator less than denominator. 

• Equivalent Selfie Fractions: Addable and Dottable Together

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for dottable fractions, i.e., instead of additions we have multiplication. In this work, we have written equivalent selfie fractions having both the operations, i.e., addition and multiplication together. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator is less than denominator. For the case of pandigital selfie fractions, only few are considered, where each representation is more than 17 times. 

•  Double Sequential Representationsof Natural Numbers - III

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations the digits used are 8 to 0 in decreasing order.

• Double Sequential Representationsof Natural Numbers - IV

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations the digits used are 9 to 0 in decreasing order. 

• Pyramidical Representations of Natural Numbers - II

This work brings natural numbers from 0 to 1000 with representations given in decreasing order in different forms written in pyramidical way 

• Flexible Power Representationsof Natural Numbers

This work brings natural numbers from 0 to 11111 written in terms of 0 to 9 in symmetrical way, with powers as permutations of same digits 0 to 9. 

• Triple Representations ofNatural Numbers - I

This work brings representations of natural numbers in three different ways. One is based on power of same digits used in bases with permutations. The other two are based on increasing and decreasing orders of digits by use of basic operations along with square-root and factorial. Number of digits in each representation are understood as width. This work is up to 6 digits or width 6. 

Taneja has 315 publications listed on his ResearchGate site. I'll probably create some posts based on his papers in the near future.

The Juggler Sequence

Today I stumbled upon the so-called Juggler sequence explained by Wikipedia as

an integer sequence that starts with a positive integer \(a_0 \), with each subsequent term in the sequence defined by the recurrence relation:

$$a_{k+1}= \begin{cases}

\left \lfloor a_k^{\frac{1}{2}} \right \rfloor, & \mbox{if } a_k \mbox{ is even} \\

\\

\left \lfloor a_k^{\frac{3}{2}} \right \rfloor, & \mbox{if } a_k \mbox{ is odd}.

\end{cases}$$

Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler. If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verified for initial terms up to one million, but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems". 

Figure 1 shows the SageMath code (permalink) to determine the trajectory for any given number, along with the numbers of steps required and the maximum value reached.

Figure 1

Most numbers reach the value 1 quickly but others are more stubborn. Records are set as we move through the natural numbers and these numbers form OEIS sequence A094679. The sequence begins:

1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, 1155, 4065, 4229, 4649, 7847, 13325, 34175, 59739, 78901, 636731, 1122603, 1301535, 2263913, 5947165, 72511173, 78641579, 125121851, 198424189, ...

OEIS A094698 shows what these records are: 

0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, 166, 193, 201, 258, 263, 268, 271, 298, 335, 340, 443, 479, 484 

Comparing the two sequences we can see that there are 73 steps required for 193 to reach 1. The maximum value reached is a rather large during the trajectory is:

6743569603489758391265376070807357156339920158784377929096419715849060516985205368792190354996630779167466266586213526771780967700267133711091446786931423291036091166608223302792047793105565012490585915410391500762927066039966992101729450252321626382793545523711387059090

With such large numbers being involved, it's better to use a logarithmic scale for viewing the trajectory of a given number. For example, the trajectory of 1003 has 15 steps with maximum value 39526058. Here is its trajectory and Figure 2 gives a graphical representation: 1003, 31765, 5661392, 2379, 116035, 39526058, 6286, 79, 702, 26, 5, 11, 36, 6, 2, 1.

Figure 2: juggler trajectory of 1003

GeeksforGeeks gives the C++, C, Java, Python, C# and PHP code to generate the juggler trajectory for any natural number input.

Anti-Magic Squares

Today I turned 25307 days old and, as is my habit, I examined the entries for this number in the Online Encyclopaedia of Integer Sequences or OEIS. I've already mentioned 25307 in my previous post as forming part of a prime quadruple consisting of 25301, 25303, 25307 and 25309. For this reason, it is mentioned in OEIS A136721 Prime quadruples: 3rd term.

However, I also came across a mention of the number in OEIS A117560 \(n(n^2-1)/2 - 1\) which at first sight seemed unremarkable. However, on reading further I discovered that the numbers in this sequence form an:

approximation for the lower bound of the "anti-magic constant" of an anti-magic square of order n. The anti-magic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. 

I was of course familiar with magic squares but I'd never heard of anti-magic squares. I was prompted to investigate further.

To quote from WolframAlpha:

An anti-magic square is an \(n×n\) array of integers from \(1\) to \(n^2\) such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966.

There are no \(2 \times 2\) or \(3 \times 3\) anti-magic squares. The first occurrence is of a \(4 \times 4 \) anti-magic square and shown below are examples of anti-magic squares from \(4 \times 4\) to \(9 \times 9\):

Source: http://www.magic-squares.net/anti-ms.htm

The formula \(0.5 \times n(n^2-1)-1 \) yields 251 in the case of \(n=8 \) but it should be remembered that this number is a lower bound. There are \( 8 \times 8\) anti-magic squares that start with a higher number. Below is a diagram of an \( 8 \times 8\) anti-magic square showing the various totals from 252 to 269 (as opposed to 251 to 268 in the example above):

In the case of 25307, the value of \(n\) is 37. So the sum of any sequences formed by adding the elements in any row, column or diagonal of this \(37 \times 37\) anti-magic square of numbers have 25307 as their lower bound.

There is a similar sequence for the upper bounds of anti-magic squares. The formula for generating the terms is:$$ \lfloor \frac{n(n^3-n-3)}{2(n-1)} \rfloor $$The OEIS is A117561.