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In this paper, we worked with magic square type palindromic primes numbers of order × , in such a way that rows, columns and principal diagonals are palprimes along with extended row of rows are also palprimes. These types of distributions are named as magic square type palprimes or palprime distributions of order ×. This work brings results for palprime distributions of orders 5 × 5, 7 × 7 and 9 × 9. This work is combination of authors three papers studied in 2017.

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 5-digits prime numbers, i.e., from 10007 to 9991. In this case there are total 8363 prime numbers. Due to high quantity of prime numbers, the results are written only in length 6 with maximum up to 7 digits repetitions. The work for the prime numbers for the 3 and 4 digits are done by author in previous works. Moreover, the fixed repetition numbers are always multiples of 3.

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 one. In this work, we brought embedding procedure for the palprimes of 3 and 5 digits. There are total 15 palprimes of 3-digits, and 93 palprimes of 5-digits. The results obtained are of multiple choices. In case of 3-digits, the palprimes are written in 6 lines, and in case of 5-digits, the palprimes are written in 5 lines. There are 17 palprimes of 5-digits those contains some of the 3-digits palprimes. The results for the 7-digits palprimes shall be given in another work.

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 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].

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 representations of 1729, a famous Hardy-Ramanujan number in different situations. These representations are with single digit, single letter, Selfie-Type, Running Expressions, Equivalent Fractions, Triangular , Fibonacci, Fixed Digits Repetitions Prime Numbers Patterns , Palindromic-Type, Polygonal-Type, Prime Numbers, Embedded, Repeated, etc. Ideas toward magic squares are also extended. Some quotes and historical notes on Ramanujan's life and work are also given. On a special day: January 29, 17 (17-29) An equation means nothing to me unless it expresses a thought of God.-S. Ramanujan Another famous quote of Ramanujan on his dreams: While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing. S. Ramanujan Confirmations to Ramanujan's dreams: Ono and his colleagues (Emory University, Atlanta, GA, USA) drew on modern mathematical tools that had not been developed before Ramanujan's death to prove this theory was correct. We proved that Ramanujan was right. We've solved the problems from his last mysterious letters. For people who work in this area of math, the problem has been open for 90 years. We found the formula explaining one of the visions that he believed came from his goddess. K. Ono Read more at: 1.

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 operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with factorial in reverse order of digits. The work is up to 5-digits numbers.

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 operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with factorial. The work is up to 5-digits numbers.

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 operations, factorial and square-root. This paper extends authors previous work for Fibonacci sequence type selfie numbers with square-root. The work is up to 5-digits numbers.

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

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 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.

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 use of basic operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, quadratic numbers, cubic numbers, etc. This paper brings selfie numbers with square-root up to seven 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 use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have re-written selfie numbers in reverse order of digits using, factorial. The work is up to 6 digits numbers. The digit's order selfie numbers with factorial can be seen in author's another work.

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 use of basis operations along with factorial, square-root, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have re-written selfie numbers in digit's order using factorial. The work is up to 6 digits numbers. The reverse order selfie numbers with factorial can be seen in authors's another work.

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 digits numbers, i.e., up to 999. Between 10 to 99 there are many numbers are not available. Between 100 to 999, only 104 is not available. This we have written in terms of positive negative coefficients. The rest work is in positive coefficients.

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. Second, expressions with Factorial and Fibonacci values. Third, expressions with Factorial and Triangular numbers. Forth, expressions with Fibonacci sequence and Triangular numbers. The operations used are addition, subtraction and multiplication.

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, second when there is a repetition of digits but only with positive sign. The third type is with repetition of digits with positive and negative signs. In all the cases the digits follow the same order but not the operations. Operations used are only addition, subtraction and multiplication. This work is a combination of author's previous two papers [18, 19].

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 four parts. For other parts refer reference list.

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 four parts. For other parts can be seen in reference list.

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. This paper bring numbers 5001 to 10000 in terms of each digit. The total work up to 20000 numbers divided in four parts.

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 squares. In case of magic square of order 10, two different ways are written. One as a general magic square without any block. Another as block-bordered magic squares with inner magic square of order 8. Again the inner magic square can be written in two ways, i.e., one just pandiagonal and another pandiagonal and bimagic. In case of single digit the representations of numbers are not uniform. Writing in terms of single letter "a", we can get uniformity in representations of numbers. It is done in another work.

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 same pythagorean triples is done. One generating them. Second, writing patterns, and third to bring pandigital-type patterns. The pandigital-type patterns are written in two different ways, one is like, 12345678987654321, and second is of type 102030405060708090807060504030201. This work is revised form of author's previous two works.