# Inder J. TanejaFederal University of Santa Catarina | UFSC · Departamento de Matemática

Inder J. Taneja

Ph.D. from Delhi University, India

## About

315

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## Publications

Publications (315)

The idea of bordered magic squares is well known in the literature. In this work, bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2021. The work is for the orders 3 to 26. The work include fractional and decimal numbers entries having positive and/or negative signs. In some cases, the s...

The idea of embedding palindromic prime (palprime) numbers in the form of pyramid or tree is very famous in the literature, where previous palprime is in the middle of next, and so on. In these situations there is no limit where it ends, because always we find next palprime containing previous one. In this work, we brought palprimes of lengths 3, 5...

This work brings prime numbers formed digits appearing in prime number. We have considered all the prime numbers from single digits to 5 digits. For higher digits refer to further works. This type of work is known in the literature but still is not aware of references in this direction. These shall be filled as long as we know them.

The author worked with patterns in prime numbers in different situations, i.e., in terms of lengths, such as 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a new prime number. These types of patterns are of fixed length. In this work, the prime patterns are written for the...

In preivous years, the author worked with patterns in prime numbers in different situations. It means, in terms of lengths, such as, 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a prime number. These types of patterns are of fixed length. In this work, the prime patterns...

In past years, the author worked with patterns in prime numbers in different situations. It means, in terms of lengths, such as 10, 9, 8, 7 and 6. The prime patterns are understood as fixed digits repetitions along with prime number resulting again in a new prime number. These types of patterns are of fixed length. In this work, the prime patterns...

The embedded palindromic prime (palprime) numbers are generally represented in the form of pyramid or tree. These types of embedded palprimes are very famous in the literature, where previous palprime is in the middle of next, and so on. In these situations there is no limit where it ends, because always we find next palprime containing previous on...

This paper brings simplified and symmetric procedure to generate Pythagorean triples. These triples are obtained in different procedures. First procedure is given in three blocks. The second procedure is the extension of first procedure, but in little different way. These triples are applied to generate perfect square sums magic squares of consecut...

The Pythagoras theorem is very famous in the literature of mathematics. The aim of this work is to extend in a symmetrical way the some Pythagorean triples resulting in patterns. These symmetric extensions are in such a way that we reach to good patterns. In some cases, the final sums also give a good pattern. In some cases examples are with intere...

The Pythagoras theorem is very famous in the literature of mathematics. It lead us to many Pythagorean triples. Some of these triples allows us to write as pattern. The aim of this work is to write patterns in Pythagorean triples in a sequential way, i.e., starting from 3 to 1000. Patterns in Pythagorean triples are also studied by author in previo...

The author in 2017, worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns . In this work, multiple choice prime patterns are considered. Multiple choice means that the same pa...

In 2017, theauthor worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns . In this work, multiple choice prime patterns are considered. Multiple choice means that the patterns...

The author [7] worked with patterns in prime numbers in such a way that they can be extended by inserting digits in repeated ways, and still remains prime numbers. These types of prime numbers are called fixed digits repetitions prime patterns. In this work, multiple choice prime patterns are considered. Multiple choice means that the same pattern...

This paper brings natural numbers from 20001 to 40000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author's works [15, 16, 17]. For more details and comments see author's site [20].

This paper brings natural numbers from 80001 to 100000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author's works [15, 16,17]. For more details and comments see author's site [20].

This paper brings natural numbers from 40001 to 60000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author's works [15, 16,17]. For more details and comments see author's site [20].

This paper brings natural numbers from 60001 to 80000 written in ascending and descending orders of 1 to 9. The numbers are obtained by using basic operations, and factorial. For previous results see author's works [15, 16,17]. For more details and comments see author's site [20].

There are many ways of writing amicable numbers. One with divisions and sums. The other with pair of powers of each other. There is another way to represent is in product. In this paper, we brings amicable numbers in pairs in terms of products and powers. The idea of self-amicable is also introduced. Few blocks of symmetrical amicable numbers multi...

Author studied many ways of writing selfie numbers (reference [13]). By selfie numbers, we understand those numbers that can be expressed in terms of same digits as of number, either in digit's order or reverse order of digits. These numbers are obtained by use of basic operations along with factorial, square-root, etc. There are numbers very much...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, work is with binomial coefficients type selfie numbers. This paper brings combine result...

In the previous papers [9, 13], the author worked with palindromic-type expressions. These expressions are by use of operations addition and multiplications. When the operations are removed, we get even order palindromes. In the previous works, it is limited to particular cases including patterns. The final sum is either palindromic or non palindro...

In the previous papers [9, 13, 14, 15], the author worked with palindromic-type expressions. These expressions are by use of operations of addition and multiplications. When the operations are removed, we get normal even order palindromes. The previous works are limited to particular cases including patterns. The final sums are either palindromic o...

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit's order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by us...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. In two variables, the selfie numbers are obtained by use of binomial coefficients, S-gonal numbers, centered polygonal numbers, etc. This paper...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. In two variables, the selfie numbers are obtained by use of binomial coefficients, S-gonal numbers, centered polygonal numbers, etc. Quadratic...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operat...

In the previous papers [9, 13], the author worked with palindromic-type expressions. These expressions are by use of operations addition and multiplications. When the operations are removed, we get normal even order palindromes. In the previous works, it is limited to particular cases including patterns. The final sum is either palindromic or non p...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations,
such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factoria...

Numbers represented by their own digits by certain operations are considered as Selfie Numbers. There are many ways of representing Selfie Numbers, such as, numbers written in digit's order or its reverse. It can also be represented in increasing and/or decreasing order of digits. This is generally obtained by use of basis operations along with fac...

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit's order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by us...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operat...

This work brings equivalent fractions for 10-digits without repetition of digits. Sometimes known as pandigital. For the 3 to 9 digits see first part of this work. For repetition of digits, the number of equivalent fractions increases too much. These shall be dealt elsewhere.

This work brings equivalent fractions without repetition of digits. The work is for two digits numerators. Sor single digits numerator see the link. For the higher digits numerators the work in given in another papers.

This work brings equivalent fractions from 3 to 9 digits without repetition of digits. For repetition of digits the the number of equivalent fractions increases too much. The 10-digits equivalent fractions are in the second part [15].

This work brings equivalent fractions without repetition of digits. This work is for single digit numerators. For the two and higher digits numerators the work in given in another papers.

This work brings equivalent fractions without repetition of digits. The work is for three digits numerators.

The addable fractions are proper fraction where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, multiplication, potentiation, etc. For more details refer author's work [10]. This work bring mixed selfie fractions with all operatio...

The numerator and denominator of a fraction represented by same digits with certain operations, we call as seflie fractions. These operations can be done by use of operations as addition, subtraction, multiplication, potentiation, etc. For example, in case of addition, let's call it as addable selfie fractions, in case of multiplications, let's cal...

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in diff...

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in diff...

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations in diff...

The addable fractions are proper fractions where addition can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for other operations, such as, addition, subtraction, multiplication, potentiation, etc. This work brings selfie fractions with single and/or multiple representations having...

By selfie fractions, we understand that a fraction, where numerator and denominators are represented by same digits, with basic operation. This paper brings patterned selfie fractions. Patterned selfie fractions are understand as selfie fractions extendable in symmetric way. This work is with repeated digits.

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called as wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit's order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by us...

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, but with different permutations. The expressions are by use of operations of addition and/or subtraction. These numbers are cal...

This paper brings numbers in such a way that both sides of the expressions are with the same digits. One side is numbers with powers, while other side just with numbers having same digits, such as, a^b+c^d+... =ab+cd+..., etc. The the expressions studied are with positive and negative coefficients. Work is done for 2 to 10 terms expressions. From 5...

This paper brings numbers in such a way that both sides of the expressions are with same digits and in same order. One side is digits with factorial and another side are with same digits with respective powers. These types of expressions, we call as selfie expressions. Three types of expressions are studied. One when digits involved are distinct, s...

his paper brings numbers in such a way that both sides of the expressions are with the same digits. One side is numbers with powers, while other side just with numbers having same digits, such as, a^b+c^d+... = ab+cd+..., etc. The the expressions studied are with positive coefficients. The work is the increasing order of numbers maximum up to 3 dig...

Selfie expressions are written in such a way that both sides of the expressions are with same digits. This work brings expressions where one side with factorial, and other side with Fibonacci and/or with triangular numbers having same digit's order. This we have done in different ways. One expressions with Factorial, Fibonacci and Triangular values...

This paper brings magic squares of orders 3 to 10 in terms of crazy representations. These representations are of three types. One in increasing order of digits starting from 1. The second in decreasing order of digits ending in 1. The third is also in decreasing orders ending in 0. These representations are neither ending or starting from 9. Minim...

This short work brings representations of 2020 in different situations. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, magic squares, etc..

This paper brings magic squares of orders 3 to 10 in terms of permutable powers-bases digits. It means that in a same numbers, there are same digits in powers and bases. In case of order 10 there are two possibilities are given. One as general normal magic squares and another as block-bordered magic square, with inner block as pandiagonal magic squ...

It is well known that every magic square can be written as perfect square sum of entries. It is always possible with odd number entries starting from 1. In case of odd order magic squares we can also write with consecutive natural number entries. Still, it is unknown whether it is possible to even order magic squares. In case of odd order magic squ...

This short paper brings representations of 2019 in different situations. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated, magic squares, etc.

This paper brings natural numbers from 11112 to 20000 written in ascending and descending orders of 1 to 9. Most of the numbers are obtained by using basic operations, except few. It is revised version of author's previous work done in 2018, where for the missing numbers factorial and square-root are used . In this work, only \textbf{factorial} is...

This paper brings natural numbers from 20001 to 30000 written inascendinganddescendingorders of 1to 9. Most of the numbers are obtained by using basic operations, except few. It is revised version of author’sprevious work [39, 40], where for the missing numbersfactorialandsquare-rootwere used. In this work, onlyfactorialare used for missing numbers...

This paper brings natural numbers from 1 to 20000 written in terms of Fibonacci sequence and Triangular numbers. The numbers are obtained just with operation of addition. This work is revised and extended version of authors previous works done in 2018.

The natural numbers form 1 to 11111 are written in terms of single letter "a" in two different ways. One is running-type expressions, and second is fraction-type expressions. In this work, we considered the fraction-type way. It means the numbers 1 to 11111 are written as fraction-type using only the single letter "a" . The single letter "a" can ha...

This work brings natural numbers from 0 to 1000 written in two different forms. One in terms of power and bases, and another in decreasing order of numbers ending in 0. These representations are of pyramid-type. This work is combined version of author's two previous works done in 2016.

In previous works [4], running equalities are written in terms of 1 to 9 and 9 to 1 or 9 to 0 separated by single or double equality signs. Each digits is used with basic operations, along with factorial, square-root and Fibonacci sequence. These types of equalities, we called as running expressions. This work brings double and triple equality type...

This work brings representations of palindromic and number patterns in terms of single letter "a". Some examples of prime patterns are also considered. Different classifications of palindromic patterns are considered, such as palindromic decomposition, double symmetric patterns, number patterns decompositions, etc. Number patterns with powers are a...

The natural numbers form 1 to 11111 are written in terms of single letter "a" in two different ways. One running-type expressions, and second fraction-type expressions. In this paper, we worked with running-type expressions. It means the numbers 1 to 11111 are written in terms of single letter "a" . The letter "a" can have any value from 1 to 9, an...

This paper works with extensions of narcissistic numbers as flexible powers narcissistic numbers with positive, and positive-negative coefficients. This is reorganized version of author's previous work [12] done in 2017.

This paper brings extensions of narcissistic numbers with division. These extensions are done in different situations, such as, with positive and negative coefficients, fixed and flexible powers. Comparison with previous known numbers are also given.

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of flexible powers u...

This paper works with representations of natural numbers from 0 to 20000 written in terms of expressions with addition, subtraction and exponents. Digits from 0 to 9 and 1 to 9 are used in such a way that for each number represents with same digits in bases and exponents with different permutations. Some numbers can be written in more than one way,...

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of flexible powers u...

This paper brings magic squares of orders 3 to 10 in terms of single letter. In case of orders 8 and 9 there are two possibilities, i.e, one as normal magic square and another as bimagicÂ square. In this situation, the magic squares are written in both the possibilities. In case of order 10, again there are also two possibilities. One without any b...

This work brings patterned representations of natural numbers from 1 to 1000 in terms of single digits from 1 to 9. To bring these results, only basic operations, such as addition, subtraction, multiplication and division are used. This work is an extension of author's previous works in patterned form. Patterns representations are some what similar...

There are different ways of representing natural numbers, such as, writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of single digit. Th...

There are different ways of representing natural numbers, such as, writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper extends the authors previous work on representation of natural numbers in terms of single digit. Th...

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 10001 to 15000 in terms of each digit. The total work up to 20000 numbers divided in f...

There are different ways of representing natural numbers, such as writing in terms of 1 to 9 or 9 to 1, writing in terms of single letter, single digit, flexible power, etc. These types of representations we call as crazy representations. This paper bring numbers 15001 to 20000 in terms of each digit. The total worEk up to 20000 numbers divided in...

In this paper we worked with generating Pythagorean triples by use of two variables Pythagoras theorem. Based on these triples pythagorean patterns are also calculated by use of same formula of two variables Pythagoras theorem. Again by use of same formula, pandigital-type pythagorean triples are obtained. In other words, three ways study on the sa...

This paper brings traditional magic squares of orders 3 to 10 in terms of single digit. In this case, the magic squares are written separately for each digit, i.e., for the digits 1 to 9. This has been done for all the orders 3 to 10. In case of orders 8 and 9 there are two possibilities, i.e, one as normal magic squares and another as bimagic squa...

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine bordered and block-wise magic squares, for the magic squares of prime and double prime orders. We cal...

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine the both, i.e., bordered and block-wise magic squares, for the magic squares of prime and double prim...

In this work, we shall brings number patterns based on magic square sums for the magic and bimagic squares constructed with different digits. In each case, as the digits increases in cells, the values of magic and bimagic squares sums also increases giving interesting number patterns. This is done for the magic squares of orders 5, 7, 8, 9 and 10.

It is well known that every magic square can be written as perfect square sum of entries. It is always possible with consecutive odd numbers entries starting from 1. In case of odd order magic squares we can also write with consecutive natural numbers entries. In case of even order magic squares it is possible with consecutive fraction numbers entr...

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine bordered and block-wise magic squares, for the magic squares of prime and double prime numbers orders...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 110. Based on these two big magic inner order magic squares multiples of 10 are studied. By inner orders we understan...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 112. Based on these two big magic inner order magic squares multiples of 8 are studied. By inner orders we understand...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 108 and 102. Based on these two big magic inner order magic squares multiples of 6 are studied. By inner orders we understand...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 140 and 126. Based on these two big magic squares, the inner order magic squares multiples of 14 are studied. By inner orders...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 108 and 104. Based on these two big magic inner order magic squares multiples of 4 are studied. By inner order we understand...

During past years author worked with block-wise, bordered and block-bordered magic squares. This work make connection between block-wise and bordered magic squares. We started with block-wise bordered magic squares of orders 120 and 108. Based on these two big magic squares, the inner order magic squares multiples of 12 are studied. By inner orders...

This short work brings 21 main representations of 2021 in different ways. These representations are of crazy-type, running numbers, single digit, single letter, Triangular, Fibonacci, palindromic-type, prime numbers, embedded, repeated digits, colored patterns, magic squares, etc.

In this work, the days of year 2020 are represented in two different ways. One way is numerical, and another way is geometrical. In numerical representations, four different forms are considered. The first one is of crazy type, and the second one is of power type. In these two cases the year ends in 20. The third one is representations in terms of...

This work brings patterned representations of natural numbers from 1 to 1000 in terms of single letter a. For any value of letter a from 1 to 9, the results are always same. To bring these results, only basic operations, such as addition, subtraction, multiplication and division are used. This work is extension of author's [11] work in patterned fo...

The idea of bordered magic squares is well known in the literature. In this work, bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2020. The work is for the orders 3 to 25. In each case, a symmetric result is found for the magic sums of sub-magic squares. The work include fractional, dec...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci sequence, Triangular numbers, etc. These operations are applied for single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operat...

Author studied many ways of writing selfie numbers [8]. There are numbers very much near to selfie-numbers, but are not selfie numbers. These types of numbers, referred as semi-selfie numbers, where numbers are written in terms of expressions with positive and negative signs having same digits on both sides of the expressions, except the power valu...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factoria...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. These are by use of single variable. In two variables, we worked with binomial coefficients type selfie numbers with basic operations, factoria...

By selfie numbers, we understand that the numbers represented by their own digits by use of certain operations, such as, basic operations, factorial, square-root, Fibonacci squence, Triangular numbers, etc. Some binomial coefficient-type patterned selfie numbers are also studied. This paper works with binomial coefficient type selfie numbers. This...