Methods of elementary Number Theory

In summary, this person is asking if it is necessary to prove the parameterization of a solution to a diophantine equation in order to enumerate all of the solutions.
  • #1
evinda
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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)
 
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  • #2
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.
 
  • #3
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 $X^2 + Y^2 + Z^2 = 0$.
 
  • #4
Bacterius said:
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 said:
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)
 
  • #5


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!
 

FAQ: Methods of elementary Number Theory

1. What is elementary number theory?

Elementary number theory is a branch of mathematics that deals with the properties and relationships of integers. It focuses on topics such as prime numbers, divisibility, and factors.

2. What are the basic methods used in elementary number theory?

Some of the basic methods used in elementary number theory include prime factorization, modular arithmetic, and the Euclidean algorithm.

3. How is elementary number theory used in cryptography?

Elementary number theory is used in cryptography to create secure codes and ciphers. This is because many of its concepts, such as prime numbers and modular arithmetic, are difficult to reverse and can be used to create strong encryption methods.

4. Can elementary number theory be applied to real-world problems?

Yes, elementary number theory has many real-world applications, including in computer science, cryptography, and coding theory. It can also be used to solve problems in fields such as physics and engineering.

5. Is elementary number theory a difficult subject to learn?

This depends on the individual and their mathematical background. Some of the concepts in elementary number theory can be challenging, but with dedication and practice, it can be a fascinating and rewarding subject to learn.

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