[seqfan] Pythagorean triangles

[michel.marcus at free.fr]

Dear all, 


Carlos Rivera's problem n°82 asks for Pythagorean triples that are 3-gonal, 4-gonal and 5-gonal in any order. 


I derived a few other problems from this. 


The (100,105,145) triple is 4,3,5. 
So for (3,4,5) and (100,105,145), we have x-gonal^2 + y-gonal^2 = z-gonal^2. 
I am using A176774 here. Are there others like this ? 


Another problem is i f you apply A176774 onto (3,4,5), you get (3,4,5). Are there others like this ? 



Another related sequence in OEIS is A177025. 
If you apply A177025 onto (3,4,5), you get 1,1,1. Are there other such triples ? 


A177028 is a sequence related to A176774. 
In A177028, we have that for instance, 105 is 3-gonal, 12-gonal, 36-gonal and 105-gonal. 
And the Pythagorean triples satisfy x-gonal^2 + y-gonal^2 = z-gonal^2 for the 1st column of A177028. 
Can we find a triple where it works for other columns ? 


The (3,3,3) triple 8778^2 + 10296^2 = 13530^2 is mentioned in Problem 81 and A213188. 
Is it possible to find a (k,k,k) triple ? with A176774 or even with A177028 ?? 


I hope these make sense. 


Cheers. 
MM