MHB Pythagorean Triples- Why is this the proof?

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Proof
AI Thread Summary
The discussion centers on the derivation of Pythagorean Triples, specifically the formula for generating them: (x,y,z) = (r²-s², 2rs, r²+s²) where r and s are coprime integers, r > s, and r is odd while s is even. The proof involves transforming the equation x² + y² = z² into a circle's equation and using rational slopes to find rational points on the circle, which leads to the identification of integer solutions. The discussion also highlights that the method produces Primitive Pythagorean Triples, attributed to Euclid but possibly known to earlier civilizations. The importance of the greatest common divisor (gcd) in ensuring the triples are primitive is emphasized, as well as the connection to complex numbers in generating these triples. Overall, the thread explores the mathematical foundations and proofs behind the generation of Pythagorean Triples.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

The solutions of $X^2+Y^2=Z^2$ are called Pythagorean Triples .

We suppose that $x$ odd and $y$ even.

All the solutions $(x,y,z)$ with $gcd(x,y,z)=1$ are given by:

$$(x,y,z)=(r^2-s^2,2rs,r^2+s^2)$$

$$r, s \in \mathbb{Z}, r>s, (r,s)=1$$

$$r \not\equiv s \pmod 2$$

Could you explain me why the following is the proof of the above sentence? (Sweating)

View attachment 3324

Line equation $AB$: $y-y_0=\lambda(x-x_0)$

so, $v=t(u+1)$.

Therefore, the point $B=(u,v)$ is a solution of the system:

$$\begin{Bmatrix}
u^2+v^2=1\\
v=t(u+1)
\end{Bmatrix}$$

Replacing $v$ at the first equation, we get:

$$(1+t^2)u^2+2t^2u+(t^2-1)=0$$

We solve the equation and find:

$$u=\frac{1-t^2}{1+t^2} , u=-1$$

$u=-1$ corresponds to the point $A=(-1,0)$.

We calculate $v=t(u+1)=\frac{2t}{t^2+1}$, so the coordinates of $B$ are:

$$B= \left ( \frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right )$$

Remark:

If the slope $t \in \mathbb{Q}$, then $B=(u,v) \in \mathbb{Q} \times \mathbb{Q}$.

Conversely, if the coordinates of $B=(u,v)$ are rational, then the line $AB$ has slope $t=\frac{v-0}{u+1} \in \mathbb{Q}$.

Therefore, the set of rational solutions of $x^2+y^2=1$ is:

$$\begin{Bmatrix}
(u,v)=\left ( \frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right )|t \in \mathbb{Q}
\end{Bmatrix} \cup \{ (-1,0)\}$$

Remark: If $t=\frac{r}{s}, r, s \in \mathbb{Z}, (r,s)=1$, we get the Pythagorean Triples.

But... how do we get them? (Sweating)
 

Attachments

  • proof.png
    proof.png
    3.7 KB · Views: 134
Last edited:
Mathematics news on Phys.org
evinda said:
Hello! (Wave)
Therefore, the set of rational solutions of $x^2+y^2=1$ is:

$$\begin{Bmatrix}
(u,v)=\left ( \frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right )|t \in \mathbb{Q}
\end{Bmatrix} \cup \{ (-1,0)\}$$

Remark: If $t=\frac{r}{s}, r, s \in \mathbb{Z}, (r,s)=1$, we get the Pythagorean Triples.

But... how do we get them? (Sweating)

When h=1
$(u,v,h)=(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2}, 1)$

For h>1
Multiply u,v and h by $1+t^2$
$(u',v',h')=(1-t^2, 2t, 1+t^2)$

Subst t=s/r
Multiply u',v' and h' by $r^2$
 
Last edited:
A standard method to solve diophantine equations is to use fundamentals of analytic geometry (later which turns out to reveal a more general connection with arithmetic geometry and algebraic geometry) Here's an explanation : recall that $x^2 + y^2 = 1$ is the analytic definition of a circle with radius $1$ situated at the origin. So all rational points $(x, y)$ must lie on the circle in the Cartesian plane. Now draw a straightline from $(-1, 0)$ to some loci $(u, v)$ of the circle, i.e., it's a point which continuously "loops" around the origin.


Recall the equation $y - y_0 = \lambda (x - x_0)$ of straightlines in Cartesian coordinates. $\lambda$ is the slope $\tan(\theta)$, which is $\overline{OA}$ here, $O = (0, 0)$ being the origin. $(x_0, y_0) = (-1, 0)$ so subbing in gives the equation $y = \lambda (x + 1)$. This must also hold at the moving point $(u, v)$ on the circle, for which $v = \lambda (u + 1)$. Thus, we have two relations between $u$ and $v$ while $(u, v)$ is moving on the path we set up for it : the standard $u^2 + v^2 = 1$ and $v = \lambda (u + 1)$. However, when $(u,v)$ stumbles onto a rational point on the circle (there are infinitely many, take $(3/5, 4/5)$), then from the second relation, $\lambda$ must be rational also (quotient of two rationals is a rational). Thus, ALL of rational points $(a, b)$ on the circle $x^2 + y^2 = 1$ satisfies

$a^2 + b^2 = 1$ AND
$a = t(b+1)$ for some *rational* $t$ (the slope).

Subbing the second into the first, $t^2 (b+1)^2 + b^2 = 1 \Longrightarrow (t^2+1)b^2 + 2bt^2 + t^2-1 = 0$ which factors as $((1+t^2)b-(1-t^2))(b+1)$. Solving for $b$ and finding $a$ from the second equation gives you the desired solution set (well, a *complete* solution set, i.e., all rational solutions are of that form).
 

Attachments

  • Capture.PNG
    Capture.PNG
    4.3 KB · Views: 106
mathbalarka said:
A standard method to solve diophantine equations is to use fundamentals of analytic geometry (later which turns out to reveal a more general connection with arithmetic geometry and algebraic geometry) Here's an explanation : recall that $x^2 + y^2 = 1$ is the analytic definition of a circle with radius $1$ situated at the origin. So all rational points $(x, y)$ must lie on the circle in the Cartesian plane. Now draw a straightline from $(-1, 0)$ to some loci $(u, v)$ of the circle, i.e., it's a point which continuously "loops" around the origin.


Recall the equation $y - y_0 = \lambda (x - x_0)$ of straightlines in Cartesian coordinates. $\lambda$ is the slope $\tan(\theta)$, which is $\overline{OA}$ here, $O = (0, 0)$ being the origin. $(x_0, y_0) = (-1, 0)$ so subbing in gives the equation $y = \lambda (x + 1)$. This must also hold at the moving point $(u, v)$ on the circle, for which $v = \lambda (u + 1)$. Thus, we have two relations between $u$ and $v$ while $(u, v)$ is moving on the path we set up for it : the standard $u^2 + v^2 = 1$ and $v = \lambda (u + 1)$. However, when $(u,v)$ stumbles onto a rational point on the circle (there are infinitely many, take $(3/5, 4/5)$), then from the second relation, $\lambda$ must be rational also (quotient of two rationals is a rational). Thus, ALL of rational points $(a, b)$ on the circle $x^2 + y^2 = 1$ satisfies

$a^2 + b^2 = 1$ AND
$a = t(b+1)$ for some *rational* $t$ (the slope).

Subbing the second into the first, $t^2 (b+1)^2 + b^2 = 1 \Longrightarrow (t^2+1)b^2 + 2bt^2 + t^2-1 = 0$ which factors as $((1+t^2)b-(1-t^2))(b+1)$. Solving for $b$ and finding $a$ from the second equation gives you the desired solution set (well, a *complete* solution set, i.e., all rational solutions are of that form).

Why do we prove in that way that all the solutions with $gcd(x,y,z)=1$ are given by:

$$(x,y,z)=(r^2-s^2,2rs,r^2+s^2)$$

$$r,s \in \mathbb{Z}, r>s , (r,s)=1, r \not \equiv 2(2)$$

? (Thinking) (Sweating) :confused:
 
Note that $a^2 + b^2 = c^2$ can be transformed into $x^2 + y^2 = 1$ where $x = a/c$ and $y = b/c$. Thus, from our previous calculations,

$$(\frac{a}c, \frac{b}c) = \left ( \frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2} \right )$$

For all $(a, b, c) \in \Bbb Z^3$ for some rational $t$. As $t$ is rational, we can sub in $t = s/r$ for integers $s, r$. That gives us

$\displaystyle \frac{1-t^2}{1+t^2} = \frac{1-s^2/r^2}{1+s^2/r^2} = \frac{1-s^2/r^2}{1+s^2/r^2} \cdot \frac{r^2}{r^2} = \frac{r^2 - s^2}{r^2 + s^2}$ and $\displaystyle \frac{2t}{1+t^2} = \frac{2s/r}{1+s^2/r^2} = \frac{2s/r}{1+s^2/r^2} \cdot \frac{r^2}{r^2} = \frac{2sr}{r^2 + s^2}$

Hence, $\displaystyle (a, b) = \left ( \frac{r^2 - s^2}{r^2 + s^2} \cdot c, \frac{2rs}{r^2 + s^2} \right )$. But $a, b$ are integers, thus $r^2 + s^2$ divides $c$, i.e., $c = d\cdot (r^2 + s^2)$ for some integer $d$. Subbing that in, we get $(a, b) = (d \cdot (r^2-s^2), d \cdot 2rs)$. Hence,

$$(a, b, c) = (d \cdot (r^2 - s^2), d \cdot 2rs, d \cdot (r^2 + s^2))$$

For all integer $a, b, c$. Apparently, if $\text{gcd}(a, b, c) = 1$ then $d = 1$ and $\text{gcd}(r, s) = 1$. Other conditions also follows likewise (left as an exercise for you).
 
evinda said:
Hello! (Wave)

The solutions of $X^2+Y^2=Z^2$ are called Pythagorean Triples .

We suppose that $x$ odd and $y$ even.

All the solutions $(x,y,z)$ with $gcd(x,y,z)=1$ are given by:
$$(x,y,z)=(r^2-s^2,2rs,r^2+s^2)$$
$$r, s \in \mathbb{Z}, r>s, (r,s)=1$$
$$r \not\equiv s \pmod 2$$

By the way:
The method that you provide (Sometimes called Euclid's Formula) produces a subset of Pythagorean Triples called Primitive Pythagorean Triples. Although it is attributed to Euclid, artifacts like Plimpton 322 hint that this formula was known to the Babylonians.

A simple way to remember Euclids formula is this...

Using complex numbers
Let z = r + si
Then x + yi = z2
(with the same restrictions on r,s as before)

SUMMARY: The square root of any Primitive Pythagorean Triple represented as x+yi (with x odd) is a complex number with integer components.
 
Primitive Pythagorean Triple (x odd) generation not using Euclid's Formula.Let a, b and c be three integers. Then,
x = a+c
y = b+c
z = a+b+c

The prime factorization of a, b, and c is critical.

a = 1 * p1a2 * p2a2 * p3a2 * ... * pna2
b = 2 * p1b2 * p2b2 * p3b2 * ... * pmb2
c = 2 * (p1a * p2a * p3a * ... * pna) * (p1b * p2b * p3b * ... * pmb)

Repeated use of the same prime is allowed, as long as a and b have no prime in common.
n=m=0 corresponds to the (3,4,5) triangle.
 
Last edited:
Back
Top