# Questions tagged [pythagorean-triples]

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

65 questions
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### Derivation of Pythagorean Triple General Solution Starting Point:

I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation: $$x^2 + y^2 = z^2,$$ where $x,y,z > 0$ are integers. I came across the ...
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### For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
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### Explain this convergence among Pythagorean triplets

Why do the ratios of successive values of integers $a$ and $c$, where $a^{2}+(a+1)^{2}=c^{2}$, appear to converge to $$\frac{a_{n+1}}{a_{n}},\frac{c_{n+1}}{c_{n}}\rightarrow3+2\sqrt{2}$$ I rigorously ...
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### Three pythagorean triples

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$
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### Is there a general formula for three Pythagorean Triangles which share an area?

The basic formula for generating a Pythagorean triangle $A^2 + B^2 = C^2$ is, $A = M^2 - N^2;\quad B = 2MN ;\quad C = M^2 + N^2$ And Wolfram Alpha gave me a solution (credited to an Enrique Zeleny)...
For our purpose,let $a,b,c$ and $x\gt2$ be natural numbers such that the positive integers $a,b$ and $c$ form a special pythagorean triple $(a,b,c)$,then it is conjectured that the following is true $... 1answer 74 views ### Is it true that$n = 2t^2-2$for even$t$is a congruent number? This post asks for$m$such that the simultaneous Pythagorean triples, $$a^2+m^2b^2 = c^2\\b^2+c^2 = d^2\tag1$$ have solutions. Will Jagy found an infinite family given by, $$m = 2t^2-2 = 0, 6, 16, ... 1answer 149 views ### Special Euler bricks and x^2(y^2-1)^2+y^2(x^2-1)^2=z^2 Define,$$P_1 := a^2+b^2\\ P_2 := a^2+c^2\\ P_3 := b^2+c^2$$Let,$$a,b,c = 2xy,\;x(y^2-1),\;y(x^2-1)$$and P_1,P_2 become squares. If we wish to make P_3 a square as well, then,$$P_3:=x^2(... 0answers 114 views ### Solving$x^4+y^4+z^4 =t^2$via elliptic curves This post had an answer that involved the equation, $$x^4+y^4+z^4 =t^2$$ Holding$y,z$constant, this is then a quartic polynomial in$x$to be made a square. Using a birational transformation, it ... 2answers 486 views ### Quadruple of Pythagorean triples with same area Can one find explicitly$a_i,b_i,c_i\in\Bbb N,i=1,2,3,4$so that $$a_i<b_i, \qquad \text{ and } \qquad a_i^2+b_i^2=c^2_i \qquad\text{for } i=1,2,3,4$$ and $$a_1b_1=a_2b_2=a_3b_3=a_4b_4, \... 5answers 4k views ### Can every perfect square exist as the sum or difference of two perfect squares? I believe this is trivial and I'm over-complicating it. But can every squared integer be expressed as the sum of two squared integers OR the difference of two squared integers? And is there a proof ... 2answers 150 views ### Solve for Rationals p,q,r Satisfying \frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1. Find all rational solutions (p,q,r) to the Diophantine equation$$\frac{2p}{1+2p-p^2}+\frac{2q}{1+2q-q^2}+\frac{2r}{1+2r-r^2}=1\,.$$At least, determine an infinite family of (p,q,r)\in\mathbb{Q}^3... 2answers 135 views ### How many integer-sided right triangles are there whose sides are combinations? How many integer-sided right triangles exist whose sides are combinations of the form \displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}? Attempt: This seems like a ... 1answer 180 views ### Cuboid nearest to a cube Cuboid nearest to a cube. While answering this question, euler bricks: way to calculate them? I noticed one result was not too far from cube shaped, and wondered if there was a more cubic cuboid.$$... 1answer 492 views ### If$2xy$is a perfect square, then$x^2+y^2$cannot be I ask a question that is probably never proved a conjecture: if$x$and$y$are two natural numbers$> 1$such that$2xy = N^2$( double their product is a perfect square ), x and y cannot be part ... 4answers 714 views ### How to descend within the “Tree of primitive Pythagorean triples”? It is well-known that the set of all primitive Pythagorean triples has the structure of an infinite ternary rooted tree. What is the exact algorithm (i.e., formula, or possibly set of three formulas) ... 1answer 120 views ### Euler's totient function applied to higher power triples I've been working my way through the mathematics presented in this question: Pythagorean triples that "survive" Euler's totient function concerning Pythagorean triples$a^2+b^2=c^2$for ... 1answer 192 views ### On$119^2+120^2=13^4\,$and$p=239$We are familiar with, $$\frac\pi4=4\arctan\tfrac15-\arctan\tfrac1{239}$$Let$p=a+b=239$and$(a,b,c,d)=(120,119,13,2).\,$Some years back, I observed this rather long list of Diophantine relations and ... 2answers 1k views ### Pythagorean Triple divisible by$5$Show that, if x, y and z are integers such that$x^2+y^2 = z^2$,then at least one of$x,y,z$is divisible by$5$. I was able to show that at least one of$x$or$y$is divisible by$2$. Can someone ... 3answers 3k views ### proof: primitive pythagorean triple, a or b has to be divisible by 3 I'm reading "A friendly introduction to number theory" and I'm stuck in this exercise, I'm mentioning this because what I need is a basic answer, all I know about primitive pythagorean triplets is ... 3answers 1k views ### Find all solutions to the equation$x^2 + 3y^2 = z^2$Find all positive integer solutions to the equation$x^2 + 3y^2 = z^2$So here's what I've done thus far: I know that if a solution exists, then there's a solution where (x,y,z) = 1, because if there ... 2answers 151 views ### solutions of$a^2+b^2=c^2$I am trying to figure the following out. If you have$a^2+b^2=c^2$and let$x=a/c$and$y=b/c$how can you show that$x=\frac{m^2-n^2}{m^2+n^2}$and$y=\frac{2mn}{m^2+n^2}$for some relatively ... 1answer 2k views ### Are there infinitely many pythagorean triples? I believe these questions are all asking different things, but: Are there infinitely many (integer) solutions to the pythagorean theorem? Is every positive integer part of a solution to the ... 3answers 848 views ### Pythagorean triples with the same c value$a^2 + b^2 = c^2$There are, Primitive Pythagorean Triples, that share the same c value. For example,$63^2 + 16^2 = 65^2$and$33 ^2 + 56^2 = 65^2$. I have been trying to figure out why the ... 1answer 149 views ### Find pairs of side integers for a given hypothenuse number so it is Pythagorean Triple I am trying to find all pairs of side integers (a, b) for a given hypothenuse number n so that (a, b, n) is a Pythagorean triple, i.e.,$ a^2 + b^2 = n^2\$ The approach i am using is Sorting the ...
For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula $$a^2 +b^2 = c^2$$ For a slightly more complicated plane; x,y and z (3 dimensional), ...