{ On Pythagorus Triples Expansion } Homepage

On a Quick Expansion Method for Pythagorus Triples

by Frank C. Fung ( 1st published in December, 2021. )

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Summary and Key Findings :

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Summary :

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Key Finding One :

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Key Finding Two :

Table of Content

Table Of Content
Part I --- Introduction and Overview
Section IIntroduction
Section IIAn Overview of the Paper
Part II --- Pythagorus Triples Basics
Section IIIFermat's { Sum of 2 Squares } Revisited
Section IVThe Usual Procedure for Expanding Pythagprus Triples
Part III --- Explaining the Quick Expansion Procedure for Pythagorus Triples
Section VOutline and Purpose for Party III
Section VISetting up the Random Triangle and its Reference Frame
Section VIICreating the 3rd-Order Polynomial Equation
Section VIIISolving the 3rd-Order Polynomial Equation
Section IXSetting up the { Value-In-Question } - [ V-sub-IQ ]
Part IV --- Numerical Examples for the Quick Expansion Procedure based on Part III
Section XOutline and Purpose for Party IV
Section XIResults for the [ 3-4-5 / 5-12-13 ] Pythagorus-Triple Pair
Section XIIResults for the [ 3-4-5 / 8-15-17 ] Pythagorus-Triple Pair
Section XIIIResults for the [ 3-4-5 / 20-21-29 ] Pythagorus-Triple Pair
Section XIVResults for the [ 3-4-5 / 12-35-37 ] Pythagorus-Triple Pair
Section XVResults for the [ 5-12-13 / 8-15-17 ] Pythagorus-Triple Pair
Section XVIResults for the [ 5-12-13 / 20-21-29 ] Pythagorus-Triple Pair
Section XVIIResults for the [ 5-12-13 / 12-15-37 ] Pythagorus-Triple Pair
Section XVIIIResults for the [ 8-15-17 / 20-21-29 ] Pythagorus-Triple Pair
Section XIXResults for the [ 8-15-17 / 20-21-29 ] Pythagorus-Triple Pair
Section XXResults for the [ 20-21-29 / 12-35-37 ] Pythagorus-Triple Pair
Section XXISummarizing on Part IV
Part V --- Doubling-Up on the 2 Pythagorus-Triple Triangles
Section XXIIOutline and Purpose for Part V
Section XXIIISetting Up the new { Triangle A-B-C } for the Doubling-Up Process
Section XXIVFirst Try at the Doubling-Up Process
Section XXVResults of Doubling-Up Process thru to the Prime Number [ 233 ]
Section XXVIDeveloping the Double-Up Expansion Formula ( I )
Section XXVIIDeveloping the Double-Up Expansion Formula ( II )
Section XXVIIIDeveloping the Double-Up Expansion Formula ( III )
Part IV --- Developing the full Expansion Formula
Section XXIXOutline and Purpose for Part VI
Section XXXSetting up the new { Triangle A-B-C } for One-On-One Expansion
Section XXXIDeveloping the One-On-One Expansion Formula ( I )
Section XXXIIDeveloping the One-On-One Expansion Formula ( II )
Section XXXIIIDeveloping the One-On-One Expansion formula ( III )
Section XXXIVDeveloping the One-On-One Expansion formula ( IV )
Section XXXVDeveloping the One-On-One Expansion Formula ( V )
Section XXXVIConfirming the One-One-One Expansion Formula
Section XXXVIIChecking Out the One-On-One Expansion Formula
Section XXXVIIIStructure of the One-On-One Expansion Formula
Part VII --- Summarizing and Concluding Remarks
Section XXXIXSome Observations on Pythagorus Triples
Section XXXXConcluding Remarks

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Original dated 2017-12-27