MHB Methods of elementary Number Theory

AI Thread Summary
The discussion focuses on solving the Diophantine equation x² + y² = z² using elementary number theory methods. It emphasizes that solving such equations involves expressing solutions in a simpler, parameterized form that allows for easy enumeration. The conversation confirms that proving the provided parameterization qualifies as a solution, though it is not the only possible parameterization. The importance of classifying and enumerating solutions is highlighted as a key aspect of solving Diophantine equations. Overall, the thread reinforces the concept that multiple parameterizations can exist for the same equation.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
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)
 
Mathematics news on Phys.org
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 $X^2 + Y^2 + Z^2 = 0$.
 
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)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top