We know that the primitive Pythagorean triples can be generated using (3,4,5) as a seed and multiplied with three matrices given in 'Tree of Pythagorean triples' wiki page.
I was wondering why is such a formulation not possible for Eisenstein triples? Does such matrices exist with a seed or is it impossible at all?
Any thoughts or ideas on this??
Tree of Eisenstein triples
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Tree of Eisenstein triples
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Re: Tree of Eisenstein triples
Good question, but I suspect the answer to this is running very close to giving away a potential solution to at least one of the problems in Project Euler, so this is probably not the appropriate forum. Sorry.
Re: Tree of Eisenstein triples
The question is very old, what do you learn about it in this time? The question is amazing...MuthuVeerappanR wrote: ↑Wed Jun 29, 2016 12:56 pm We know that the primitive Pythagorean triples can be generated using (3,4,5) as a seed and multiplied with three matrices given in 'Tree of Pythagorean triples' wiki page...
If I understand, the grace of the Py triples is the conservation of the proportions (or Inradius) with the prime numbers and therefore, the non-repetition of its elements."A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1)"
In the case of 60° Eisensteins (e60) we have two primes and one pair like the Pythagorean, but in the case of 120° Eisensteins only prime numbers. The equations are very different and obviously have different properties.
it is a very interesting info in this paper: http://ijmcs.future-in-tech.net/14.3/R-Cotronei.pdf
Re: Tree of Eisenstein triples
Or, indeed, using (0, 1, 1) as a seed, although the first layer of expansion is inelegant. (This seed is an eigenvector of one of the matrices, and the other two give the same expansion).MuthuVeerappanR wrote: ↑Wed Jun 29, 2016 12:56 pm We know that the primitive Pythagorean triples can be generated using (3,4,5) as a seed
(0, 1, 1) similarly serves as a seed for the Eisenstein triples, but the number of matrices required is not three. Bearing in mind sjhillier's reply, filling in the details is left as an exercise.