From Newsgroup: sci.math

In this article we have created the table that classifies all primitive triples, shown some properties of basic triples and discussed about the
use of primitive Pythagorean triples in cryptography.

We continue our fascinating study of Pythagorean triples that we have
begun in -2 Classification of Pythagorean triples and reflection on
FermatrCOs last theorem-+ and -2Parabolic patterns in the scatter plot of Pythagorean triples-+. There are two categories of Pythagorean triples: primitive triples and multiple triples. In -2 Classification of
Pythagorean triples and reflection on FermatrCOs last theorem-+ we have classified all Pythagorean triples in a 3D table. The first page of this
table is shown in Table 1. All the other pages equal the first page
multiplied by an integer k and thus, contain only multiple triples. So,
Table 1 has the following property:

Property 1: Table 1 contains all primitive Pythagorean triples.

Table 1 contains also multiple triples. So, the triples in Table 1 are
called basic Pythagorean triples. In Table 1 the primitive triples are
colored in blue and the multiple triples in red. We see that the second
and fourth lines are multiple triples, the diagonal also contains only multiple triples. This suggests that we could remove all multiple triples
from Table 1 to create a table that contains only primitive triples.

For more detail, please read -2 Classification of primitive Pythagorean triples -+ https://pengkuanonmaths.blogspot.com/2025/09/classification-of-primitive-pythagorean.html

https://www.academia.edu/143986307/Classification_of_primitive_Pythagorean_triples


rCarCa

Discussion
In this article we have created the table that classifies all primitive triples. For doing so, we have started by creating the table that
classifies basic Pythagorean triples. Then, we have removed all the lines
with even index. From the resulting table we have removed all the sparse multiple triples. The remaining table is a table that classifies all
possible primitive triples.

We have discussed three properties of basic Pythagorean triples. For
example, two primitive triples can share a same value of Z. Also, sparse multiple triples are rather evenly distributed on large scale.

Finally, I have pondered about whether we can find a practical use for the classified primitive triple table. One possible use could be in
cryptography. We have shown that the Z component of a primitive triple can
be easily generated from two parameters, b and j. But we cannot easily
work out the original value of b and j from a given Z because we have two unknowns, b and j, for only one equation, Z=(b+j)^2+b^2. For example, what
are the values of b and j that generate the primitive Pythagorean triple (X=14719726793rCa, Y=518663627978rCa, Z=518872459612805)?

The heart of the RSA cryptosystem is to find two prime integers p and q so large that the factoring of the product p-+q is very hard. With large primitive Pythagorean triples, it is also very hard to work out the
original b and j from a given Z. So, I think that the three integers b, j
and Z could work at the place of p, q and p-+q for an alternate
cryptosystem similar to RSA. In addition, large prime integers are
difficult to find while b and i, with j=2i-1, can be any integers, which
would make the alternate cryptosystem simpler and cheaper than the RSA cryptosystem. The fact that the integers X, Y and Z are coprime could be useful for cryptography too.

But I am not a cryptographer and cannot evaluate the validity of this
idea. Would any cryptographer be interested?

For more detail, please read -2 Classification of primitive Pythagorean triples -+ https://pengkuanonmaths.blogspot.com/2025/09/classification-of-primitive-pythagorean.html

https://www.academia.edu/143986307/Classification_of_primitive_Pythagorean_triples



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