Double-Up Formula ( III )

On a Quick Expansion Method for Pythagorus Triples

Section XXVIII - Developing the Double-Up Expansion Formula ( III )

In this Section , we shall determine whether or not :
- [ P-Square + Q-Squre ] is a perfect square .

Let us bring back from the last Section the expression for [ P ] :
And we take note here again of the governing Pythagorus-Triple Equation :
Therefore , we can re-write the above expression for [ P ] in this new format :
Let us now do the calculations for [ P-Square ] and we have :
And we take note here that :
- yielding :
Again we take note of the governing Pythagorus-Triple Equation :
Applying this therein to the equation immediately preceeding then yields us this :
Consequently , the original expression for [ P-Square ] above , i.e. this equation :
- would now become :
And on further expansion , we have :
And the final expression for [ P-Square ] is then :

Let us bring back from the last Section the expression for [ Q ] :
Again we take note of the governing Pythagorus-Triple Equation :
As such , we can now re-write the expression for [ Q ] in this new format :
Let us now do the calculations for [ Q-Square ] and we have :
And further manipulations then yield us this final expression for [ Q-Square ] :

Let us bring back from above these expressions for [ P ] and [ Q ] :
- and
Consequently , we have this expression for [ P-Square + Q-Square ] :
- yielding :
And the final expression for [ P-Square + Q-Square ] is then :
It should be obvious that [ P-Square + Q-Square ] here is a perfect square .