Integer solutions to a^2+b^2=c^3

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?

Re: Integer solutions to a^2+b^2=c^3

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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Re: Integer solutions to a^2+b^2=c^3

Quote:

Originally Posted by

**linshi** 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?

A pairwise coprime solution