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)

 

[CellCoree]

Hello! As a scientist familiar with the methods of elementary Number Theory, I would recommend first reviewing the fundamental concepts and principles of this field before attempting to solve the diophantine equation given. This will help you better understand the problem and the techniques that can be applied to solve it.

In general, the study of diophantine equations, which are equations with integer solutions, falls under the branch of Number Theory. There are various methods and theorems that can be used to solve these types of equations, such as the Pythagorean theorem, modular arithmetic, and the Euclidean algorithm.

In the case of the diophantine equation $x^2+y^2=z^2$, one approach would be to use the Pythagorean theorem and consider the equation in terms of right triangles with integer side lengths. This can lead to the discovery of the primitive Pythagorean triples, which are solutions to the equation in its simplest form.

Another approach would be to use modular arithmetic and consider the equation in terms of remainders when divided by certain numbers. This can help identify patterns and properties of the solutions.

Once you have a good understanding of the concepts and techniques, you can then attempt to solve the equation using the given formulas. However, it is important to remember that there may be multiple solutions and it is always a good idea to check your work and verify the solutions.

I hope this helps in your exploration of elementary Number Theory and solving the diophantine equation given. Best of luck!