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 « Classification of Pythagorean triples and reflection on Fermat’s last theorem» and «Parabolic patterns in the scatter plot of Pythagorean triples». There are two categories of Pythagorean triples: primitive triples and multiple triples. In « Classification of Pythagorean triples and reflection on Fermat’s 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 « 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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1. 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, . For example, what are the values of b and j that generate the primitive Pythagorean triple (X=14719726793…, Y=518663627978…, 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 « 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