Methods of elementary Number Theory

[evinda]

Hi! (Cool)

I am given the following exercise:Try to solve the diophantine equation $x^2+y^2=z^2$ , using methods of elementary Number Theory.

So, do I have to write the proof of the theorem:

The non-trivial solutions of $x^2+y^2=z^2$ are given by the formulas:

$$x=\pm d(u^2-v^2), y=\pm 2duv, z=\pm d(u^2+v^2)$$

or

$$x=\pm d2uv, y=\pm d(u^2-v^2), z=\pm d(u^2+v^2)$$

? (Thinking)

 

[Nono713]

Yes. That's what solving an equation means, express its solutions in a simpler form, preferably one where all the solutions can be classified and easily enumerated.

 

[mathbalarka]

Given a diophantine equation $P(X_1, X_2, \cdots, X_n) = 0$ over $\Bbb Q$, "solving" it means "enumerate the solutions". Now if the zero locus (the solution set) is (countably) infinite then enumeration is essentially done by parameterization, i.e., producing a set $\{(T_1, T_2, \cdots, T_k) \in \Bbb Z^k : X_i = F_i(T_1, T_2, \cdots, T_k) \, \forall i < n\}$ for some function $F_i$ which maps integers to integers when restricted to $\Bbb Z$.

So yes, proving the parameterization you mentioned would also qualify as "solving". But it is absolutely not nessesary that this is a unique parameterization -- there are a lot of ways to completely parameterize $X^2 + Y^2 + Z^2 = 0$.

 

[evinda]

Yes. That's what solving an equation means, express its solutions in a simpler form, preferably one where all the solutions can be classified and easily enumerated.

Given a diophantine equation $P(X_1, X_2, \cdots, X_n) = 0$ over $\Bbb Q$, "solving" it means "enumerate the solutions". Now if the zero locus (the solution set) is (countably) infinite then enumeration is essentially done by parameterization, i.e., producing a set $\{(T_1, T_2, \cdots, T_k) \in \Bbb Z^k : X_i = F_i(T_1, T_2, \cdots, T_k) \, \forall i < n\}$ for some function $F_i$ which maps integers to integers when restricted to $\Bbb Z$.

So yes, proving the parameterization you mentioned would also qualify as "solving". But it is absolutely not nessesary that this is a unique parameterization -- there are a lot of ways to completely parameterize $X^2 + Y^2 + Z^2 = 0$.

Nice, thanks a lot! (Smile)