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 array in ascending order Finding the square of each element in array for(j->0 to […]

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 prime numbers $m,n \in \mathbb{Z}$

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. $$x^2+y^2=u^2$$ $$y^2+z^2=v^2$$ $$x^2+z^2=w^2$$ $x,y,z,u,v,w$ positive integers, and $x<y<z$ The result I noticed was $(240,252,275)$, and decided to use $\alpha=\large \frac{z^2}{xy}$ […]

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 three triangles which share a common area (calculated as $\frac{AB}{2}$), hence, $$M_1 N_1 (M_1^2-N_1^2)=M_2 […]

Extended Pythagorean Theorem We all well familiar with the basic Pythagorean statement – I will use a different description that will serve the discussion that follows – Given two points in $2D$ space, $(a,0)$ and $(0,b)$ with distance $d$ between them, than the relation between $a, b,$ and $d$ follows $a^2+b^2=d^2$ In $3D$ this is […]

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = y^2\tag1$$ is solvable in the rationals. Assume $x=(p/q)^2$. Then $(1)$ becomes, $$\frac{p^2}{q^6}(p^4-n^2q^4) = y^2$$ or simply, $$(p^2+nq^2)(p^2-nq^2) = w^2\tag2$$ Assuming $w=z\,t$ and equating factors, […]

This specific quadratic came up as part of a puzzle, but the context isn’t really important. I just need to find all positive integers $n$, where $5n^2+14n+1$ is a perfect square. Unfortunately I’m not really a number theorist and I don’t know enough “tricks” to make this work out. The only tricks I know are […]

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) by which one can take a given Pythagorean triple $(a,b,c)$ and find the immediately smaller triple in the tree? For example, given […]

Are there any solutions for $a, b, c$ such that: $$a, b, c \in \Bbb N_1$$ $$\sqrt{a^2+(b+c)^2} \in \Bbb N_1$$ $$\sqrt{b^2+(a+c)^2} \in \Bbb N_1$$ $$\sqrt{c^2+(a+b)^2} \in \Bbb N_1$$

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 $$a_1b_1=a_2b_2=a_3b_3=a_4b_4, \qquad c_1<c_2<c_3<c_4.$$ Some context: A Pythagorean triple is a triple $(a,b,c)\in\Bbb N$ so that $a^2+b^2=c^2$. We say that $(a,b,c)$ is primitive if $a,b$ and $c$ are coprime. In the dedicated wikipedia article the following […]

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