One solution set is:
(and removing those where b<a)
Nor is it completely unmotivated. Consider the Gaussian integers:
Suppose
a^2+b^2=c^3, a,b,c are positive integers, and b>=a.
Just by inspection, a=b=c=2 is one set of solution.
Also, a=26, b=18, c=10; a=1358, b=594, c=130; are two other sets of solutions.
If you multiply a and b by n^3 and c by n^2 for any positive integers n, you can easily get related sets of solutions.
So my the question is:
- Do other solutions exists?
- Are there any solutions, such that a,b,c are pairwise coprime, exists?
And of course I wonder how she looks like a formula describing their solutions. For the special case when
You can get a basic formula. Has the solutions:
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- What are some integers any sign. After substituting the numbers and get a result it will be necessary to divide by the greatest common divisor. This is to obtain the primitive solutions.
In the equation: If the ratio is such that the root of an integer: Then the solution is:
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And more.
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