Pythagorean Triplets In Numeric Order


First 8 generations (named)           the 9th generation (named)           


Eight Pythagorean Triplet Days to Celebrate Each Year (Like we do Pi Day) 

COOL FATHER'S DAY T-SHIRT IDEA


What if you take the inverse matrices of Hall's formulas and use them on 3,4,5?

Plotting Hall's formulas as planes in 3D?  Can you notice anything special?

Primitive Pythagorean Triplet Article

What is unique about this page is (to my knowledge) no one has thought of Primitive Pythagorean Triplets as being a family with the triplet 3,4,5 as the Adam & Eve of the 1st generation.  An advantage of this family way is how the generations of children can be classified as how they were generated.  Explanation of what a Pythagorean Triplet is and then what a Primitive Pythagorean Triplet is (kind of the prime numbers of Pythagorean Triplets) follows:

Click here for PDF version of the 8 generations of Primitive Pythagorean Triplets with explanation.  The 9th generation below.


 9 Generations of Primitive Pythagorean Triplets
Explanation: A Pythagorean Triplet is a group of three whole numbers that satisfy the Pythagorean Theorem of Right Triangles. The Pythagorean Theorem states that the lengths of the sides of any right triangle will always satisfy the condition that the sum of the squares of the legs of a right triangle will equal the square its hypotenuse. A primitive Pythagorean triplet is a root one that is not the multiple of another. Example: If 3,4,5 is a Pythagorean triplet then all multiple of 3,4,5 like 6,8,10 (multiplying by 2) or 9,12,15 (multiplying by 3) or 30,40,50 (multiplying by 10) will also be a Pythagorean Triplet however only 3,4,5 would be a primitive one.
My goal is to compile a full nine generations of Primitive Pythagorean Triplets using Hall’s UAD tree way of generating them. Explanation: Hall stated that if one has a Pythagorean triplet one can generate three more by using these nine formulas. If a & b are the lengths of the legs of the right triangle and h is the hypotenuse of a given Pythagorean Triplet then he had a go UP to formula to generate (what I call the first offspring) UP triplet of the original, a go ALONG to triplet formula in order to generate what I call a second offspring and a go DOWN formula to in order to generate the third and last offspring of the original Pythagorean Triplet.

Here are Hall’s formulas: a,b,h

go UP to (U)

a-2b+2h                       2a-b+2h                    2a-2b+3h

go ALONG to (A)

a+2b+2h                      2a+b+2h                   2a+2b+3h

go DOWN to (D)

-a+2b+2h                    -2a+b+2h                  -2a+2b+3h


then take each of these three new ones to get three more using the same formulas and so on and so on ….

so you start out with 1 Primitive Pythagorean Triplet in the first generation, use Hall’s Formulas to generate 3 more for the 2nd generation, then use Hall’s formulas go generate three for each of the 2nd generation for a total of 9 family members in the 3rd generation, 27 members in the 4th generation, 81 for the 5th generation, 243 for the 6th generation, 729 children for the 7th generation, then 2187 PPT's for the 8th generation, and finally 6,561 primitive Pythagorean Triplets in the 9th generation alone for a total of 9,831 PPT’s all listed by generation with each named with (U)P, (A)LONG, and (D)OWN representing how they came to be.  Thus (taking one randomly from the list) the Primitive Pythagorean     14523    4636   15245 UAAADD14 means this triplet (like all of them) started with 3, 4, 5 and then used Hall's Up formula to generate a 2nd triplet then his Along formula on this triplet to get a third one and then Along again to get a 4th one, Along formula again for a 5th, then the Down formula for the 6th and finally the Down formula to arrive at this 7th generation child.


Here are the generations:

Starting with the Adam & Eve of all Primitive Pythagorean Triplets

1st generation         3           4            5




2nd generation   THE 3 CHILDREN 

 

     going UP  5    12    13  U1       going ALONG  21   20     29   A2        going DOWN  15   8   17  D3

3rd generation       THE 9 GRANDCHILDREN
   7  24  25  UU1                                      55  48   73 UA2                      45    28    53  UD3

  39   80   89    AU4                                119  120  169  AA5                  77   36     85  AD6

  33   56   65   DU7                                 65    72     97   DA8                   35    12     37  DD9


4th generation THE 27 GREAT GRANDCHILDREN through the 8th generation (click here) and finally the goal was met when the 9th generation was both generated and named (click here).  (NOTE: Simply naming the 9th generation took 30 + hours.)



During the analysis of the 9 generations of Primitive Pythagorean Triplets all the 1 digit and all the 2 digits ones were found in addition to those with a 100 through a 120 were found and the multiples.

​Here is a table of all the SINGLE & DOUBLE DIGIT AND 100-120 PYTHAGOREAN TRIPLETS.

We celebrate (recognize) PI day so why not celebrate (recognize) Pythagorean Triplet Days?  After all, it is only the  RIGHT thing to do!