# Questions tagged [pythagorean-triples]

Questions about Pythagorean triples, positive integer solutions to $a^2 + b^2 = c^2$.

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### Generating Pythagorean triples using reduced fractions

I know that this question has answers here, but I'm looking for a way to make the following argument work. Let $(a,b,c)$ be a primitive Pythagorean triple with $a$ even. The goal is to prove that we ...
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(not including $0^2 + b^2$ and only including odd numbers:) Starting at $5 = 1^2 + 2^2$ then $13 = 2^2 + 3^2$ $17 = 4^2 + 1^2$ $25 = 3^2 + 4^2$ $29 = 2^2 + 5^2$ What is the equation/s to find the $\... 2 votes 1 answer 117 views ### Is the angle of the 345 triangle (pythagorean triple) related to the Geometric Progression. 4, 2, 1, 1/2, 1/4? I realised recently I could use this equation for the pythagorean triples. As you can see the 345 triangle seems to be special? as we can create it using x=2 AND x=3. I know the 345 triangle is ... 5 votes 3 answers 149 views ### Is the set$\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Phythagorean triple with}\ a<b<c\ \right\}\ $dense in$\ [0,1]\ ?$Is the set$\ \left\{ \frac{b-a }{c-a}:\ (a,b,c)\ \text{is a primitive Pythagorean triple with}\ a<b<c\ \right\}\ $dense in$\ [0,1]\ $and how do you show this? It seems likely true based on ... • 12.2k 0 votes 2 answers 71 views ### Hypotenuses for which there exist exactly 4 distinct integer triangles with an extra constraint A084648 of the OEIS contains all numbers where the square of the number can be decomposed exactly in four different ways in a sum of two squares of integers. For example 65 is a term of A084648 ... • 530 2 votes 2 answers 141 views ### Three circles internaly tangent to an equilateral triangle The diagram shows an equilateral triangle of side length 1 with 3 identical circles. Find the radius of the circle. The correct answer for the length of the right triangle in red should be$\sqrt{3}r$... • 549 2 votes 1 answer 73 views ### Generating Pythagorean triples$(a,b,c)$such that$b>a+n$for integer$n$, and$a+b$is minimum I'm trying to generate a Pythagorean triple, that is,$a,b,c \in \mathbb N$that $$a^2 + b^2 = c^2$$ under the condition that$b > a + n$,$n \in \mathbb N$and$a+b$is minimum. I tried expanding ... • 207 9 votes 2 answers 198 views ### For integers$x<y<z\$, why are these cases impossible for Mengoli's Six-Square Problem?

I got inspired by this question "Four squares such that the difference of any two is a square?" and rewrote zwim's program that is provided by his answer to the question "Solutions to a ...
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