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For $a^2+b^2=c^2$ such that $a, b, c \in \mathbb{Z}$

Do we know whether the solution is finite or infinite for $a, b, c \in \mathbb{Z}$?

We know $a=3, b=4, c=5$ is one of the solutions.

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    Look up "Pythagorean triples".2012-12-24
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    http://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple2012-12-24
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    I'm guessing that you mean "Do we know whether the number of solutions is finite or infinite?" rather than "Do we know whether the solution is finite or infinite?".2012-12-24

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Assuming $m,n$ be any two positive integers such that $m < n$, we have:

$$a = n^2 - m^2,\;\; b = 2mn,\;\;c = n^2 + m^2$$

And then $a^2+b^2=c^2$.

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    Small comment: To produce **all** triples, we need to allow the $a$, $b$, $c$ above to be multiplied by an integer constant $k$.2012-12-24
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    nice observation!2012-12-24
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To comment on whether the solution is finite or infinite note that if $(a,b,c)$ satisfy $a^2 + b^2 = c^2$, then so does $(ka,kb,kc)$ where $k \in \mathbb{Z}$. Hence, either no integer solution exists or infinite integer solution exists.

You have already observed that $(3,4,5)$ satisfies $3^2 + 4^2 = 5^2$. Hence, infinite solutions exist.

Babak Sorouh has given the parameterization, which generates almost all possible solutions $a,b,c \in \mathbb{Z}^+$ (without scaling).

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    Thanks for noting an important comment in the first paragraph. Thanks @Andre.2012-12-24
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You can use your reasoning and conclude that there are infinite right triangles that you could draw, but that is not a good proof.

It is already known that $3^2 + 4^2 = 5^2.$ You can use this statement itself to prove that there are infinitely many Pythagorean Triplets.

Let's prove that there are infinitely many Pythagorean Triplets in the form $(3k)^2 + (4k)^2 = (5k)^2,\ \, k \in \mathbb{N}$.

(NOTE that you can apply this to any Pythagorean Triplet; it's a property).

We have $(3k)^2 + (4k)^2 = (5k)^2$, or $9k^2 + 16k^2 = 25k^2$. Divide both sides by $k^2$ (it is not zero). We have $9 + 16 = 25 \iff 25 = 25$. We just have proved that the statement is true for any $k \in \mathbb{N}$ and there are infinite elements in $\mathbb{N}$.

REMARK The general proof is related.

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    I'm sorry, but I don't see how FLT tells us that the sets of solutions to $a+b=c$ and $a^2+b^2=c^2$ are infinite. Could you explain this?2012-12-24
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    Yes, according to [the context of FMT](http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Pythagorean_triples). I have edited the post to avoid such further confusions. :)2012-12-24
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    I understand that you could say these equations are related, but I don't think that Wikipedia page says anything about FLT implying that the sets of solutions for those equations are infinite (they state that they have an infinite number of solutions, but not that is an indirect or direct consequence of FLT.)2012-12-24
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    Might as well remove the Fermat's part. :\2012-12-24