Generating Pythagorean Triples, Palindromic-Type Pandigital Patterns, and Magic Squares

  • Generating Pythagorean Triples

This work brings simplified way to generate Pythagorean triples. These triples are then applied to generate perfect square sums magic squares of consecutive odd numbers, and patterned Palindromic-Type Pandigital Pythagorean Triples. These results can be in blocks of 9, 99, 999, 9999, 99999, etc. Let’s see how we proceed with this procedure. Let’s consider following three tables:

Pyth-100-1
Pyth-10000-1
Pyth-1000000-1

First table generates 9 blocks, second table generates 99 blocks, and third table generates 999 blocks of Pythagorean triples. The right hand sides of the above three tables are formed by two sums, where the last 2, 4 and  6 digits are perfect squares.  Consider a difference of squares between the terms of r.h.s. with 1 in the front of perfect square term. This difference is again a perfect squares multiples of 20, 200, 2000 etc. generating Pythagorean triples. See below:

Pyth-100-2
Pyth-10000-2
Pyth-1000000-2

  • Generating Palindromic-Type Pandigital Patterns and Magic Squares

Above three tables of Pythagorean triples again generate Palindromic-Type Pandigital Patterns and Magic Squares. The first table generates 9 blocks, the second table generates 99 blocks, the third table generates 999 blocks, and so on. In case of magic square these are 7, 97, 997, etc. For details see the link below:

I.J TANEJA – Generating Pythagorean Triples, Palindromic-Type Pandigital Patterns, and Magic Squares

Below are examples of first 9 blocks based on first table (in each case). Out of these 9 blocks, we have only 7 magic square of Orders 9 to 3. 

PG1PG2PG3PG4PG5PG6PG7PG8

For more work on magic Squares generated by Pythagorean triples – see the link below:

  1. Pythagorean Triples and Perfect Square Sum Magic Squares
  2. Palindromic-Type Pandigital Patterns in Pythagorean Triples – I

Also see the following work on Pythagorean triples:

  1. I.J. Taneja – Multiple-Type Pythagorean Patterns
  2. I.J. Taneja, Patterns in Pythagorean Triples Using Single Variable Procedures
  3. I.J. Taneja, Patterns in Pythagorean Triples Using Double Variable Procedures

 

 

 

 

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