## 31st Day of Year: 31.01.19

References: Patterns in Prime Numbers: Fixed Digits Repetitions; Single Letter Representations of Numbers

References: Patterns in Prime Numbers: Fixed Digits Repetitions; Single Letter Representations of Numbers

References: Patterns in Prime Numbers: Fixed Digits Repetitions; Pyramidical-Type Representations of Natural Numbers

References: Patterns in Prime Numbers: Fixed Digits Repetitions; Crazy Representations of Natural Numbers – The 10958 Problem

References: Same Digits Equality Expressions Pyramidical-Type Representations of Natural Numbers.

References: 1. Multiple-Type Pythagorean Patterns; 2. Pyramidical-Type Representations of Natural Numbers.

References Pyramidical-Type Representations of Natural Numbers Crazy Representations of Natural Numbers – The 10958 Problem

References Multiple-Type Pythagorean Triples Patterns Crazy Representations of Natural Numbers – The 10958 Problem

References: Patterns in Prime Numbers: Fixed Digits Repetitions ; Single Letter Representations of Numbers from 1 to 20000

References: Pythagorean Triples Generating Pandigial Palindromic-Type Patterns and Magic Squares Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, , January 19, 2019, http://doi.org/10.5281/zenodo.2544527 Inder J. Taneja, Palindromic-Type Pandigital Patterns in Pythagorean Triples, Zenodo, January 20, 2019,http://doi.org/10.5281/zenodo.2544551

Reference: I.J. TANEJA, Same Digits Embedded Palprimes, RGMIA Research Report Collection, 21(2018), Art. 75, pp. 1-47.

References Pythagorean Triples Generating Pandigial Palindromic-Type Patterns and Magic Squares Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, , January 19, 2019, http://doi.org/10.5281/zenodo.2544527 Inder J. Taneja, Palindromic-Type Pandigital Patterns in Pythagorean Triples, Zenodo, January 20, 2019,http://doi.org/10.5281/zenodo.2544551 Factorial-Power Selfie Expressions https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/

This paper brings numbers in such a way that both sides of the expressions are with same digits. 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 […]

References Pythagorean Triples Generating Pandigial Palindromic-Type Patterns and Magic Squares Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, , January 19, 2019, http://doi.org/10.5281/zenodo.2544527 Inder J. Taneja, Palindromic-Type Pandigital Patterns in Pythagorean Triples, Zenodo, January 20, 2019, http://doi.org/10.5281/zenodo.2544551 https://inderjtaneja.com/2017/02/23/same-digits-equality-expressions/ https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/

References Pythagorean Triples Generating Pandigial Palindromic-Type Patterns and Magic Squares Inder J. Taneja, Multiple-Type Patterns and Pythagorean Triples, Zenodo, , January 19, 2019, http://doi.org/10.5281/zenodo.2544527 Inder J. Taneja, Palindromic-Type Pandigital Patterns in Pythagorean Triples, Zenodo, January 20, 2019, http://doi.org/10.5281/zenodo.2544551 https://inderjtaneja.com/2017/02/23/same-digits-equality-expressions/ https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/

References https://inderjtaneja.com/2017/02/23/same-digits-equality-expressions/ https://inderjtaneja.com/2019/01/18/triangular-numbers-and-running-expressions-with-equalities/ https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/

In previous work ( Part I, Part II, Part III), 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 once with basic operations, along with factorial, square-root and […]

References https://inderjtaneja.com/2017/02/23/same-digits-equality-expressions/ https://inderjtaneja.com/2017/05/24/running-expressions-increasing-and-decreasing-orders/ https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/

References https://inderjtaneja.com/2018/09/24/pythagorean-triples-generating-pandigial-palindromic-type-patterns-and-magic-squares/ https://inderjtaneja.com/2018/11/16/crazy-representations-of-natural-numbers-the-10958-problem/