If value of $a+b+c$ is given as $n$ and it is provided that a2+b2 = c2. So, I need to find the value of $(abc)$ in terms of $n$.
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https://projecteuler.net/problem=9 @MarkBennet – Abhinav Kushagra Aug 09 '17 at 08:50
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I have already solved it and I am just curious about the direct relation. – Abhinav Kushagra Aug 09 '17 at 08:51
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1Possible duplicate of Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$ – Dietrich Burde Aug 09 '17 at 08:52
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But have you tried squaring $a+b+c=n$ and eliminating $a^2+b^2$ and $a+b$? Or anything simple like that, to make progress. – Mark Bennet Aug 09 '17 at 08:56
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If you've already solved it, show us; we can't read minds. – Shaun Aug 09 '17 at 09:52
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@Shaun By that I mean, I have solved Project Euler problem #9 by finding individual values of a, b and c. Do I need to show you my solution? – Abhinav Kushagra Aug 09 '17 at 10:08
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You can't hope for an answer in general, because two equations in three variables will generally give a one dimensional set of points. But the answer can sometimes be given in terms of a single parameter. Here it is easy enough to work through:
$$a+b+c=n$$
$$a^2+b^2+c^2+2c(a+b)+2ab=n^2$$
$$2c^2+2c(n-c)+2ab=n^2$$
$$2cn+2ab=n^2$$
$$2c^2n+2abc=cn^2$$
$$abc=\frac{cn(n-2c)}2$$
Mark Bennet
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the points $(x,y,z)$ with $x^2+y^2=z^2$ form an infinite cone in the $x,y,z$ plane.
The points $(x,y,z)$ with $x+y+z=\alpha$ form a plane. The intersection of the plane and the cone is a conic.
This conic can contain a large number of lattice points.
Asinomás
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