# Tagged Questions

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### How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
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### Diophantine quartic equation in four variables, part deux

A recent Question asked for all positive integer solutions of a simple quartic in four unknowns: $$wxyz = (w+x+y+z)^2 \tag{1}$$ whose satisfaction is necessary for the integer side lengths ...
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### Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
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### When is the area of a triangle whose side lengths are consecutive integers also an integer?

Consider a triangle with side lengths 3, 4, and 5. By Heron's formula, its area is $\sqrt{6(6 - 5)(6-4)(6 - 3)} = \sqrt{6(1)(2)(3)} = \sqrt{36} = 6$. Are there any other triangles like this?
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### Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
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### Finding integer coordinates on a sphere's surface [duplicate]

Possible Duplicate: Integer coordinate set of points that is a member of sphere surface Assume $C$ is a sphere with radius $r$ and center in the origin (0,0,0). How can we find the set of ...
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### rational triangles and cosines

I've recently started to try working on exercises from this book on Diophantine equations before I need to return it to the library. This one has me slightly stumped. It asks to show that the cosine ...
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### Heronian triangles

How to prove that all Heronian triangles can be found using formulas described here? I understand that the described substitution will give Heronian triangle, but how to prove that using the ...
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### Heronian triangle Generator

I'm trouble shooting my code I wrote to generate all Heronian Triangles (triangle with integer sides and integer area). I'm using the following algorithm $$a=n(m^{2}+k^{2})$$ $$b=m(n^{2}+k^{2})$$ ...
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### Integer coordinate set of points that is a member of sphere surface

I have a graphic application to develop which involve many spheres. I should determine then on run time. Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere ...
### Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$?
Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$? I'm fairly certain it does, but I'm hoping to see a rigorous proof of this statement. Thanks. Here is my motivation. I'm ...