Find all Pythagorean triples $x^2+y^2=z^2$ where $x=21$

Question

Consider the following theorem:

If $(x,y,z)$ are the lengths of a Primitive Pythagorean triangle, then $$x = r^2-s^2$$ $$y = 2rs$$ $$z = r^2+z^2$$ where $\gcd(r,s) = 1$ and $r,s$ are of opposite parity.

According to the previous theorem,My try is the following:

since $x = r^2-s^2$, $x$ is difference of two squares implying that $x \equiv 0 \pmod 4$. But $x=21 \not \equiv 0 \pmod 4$. Hence, there are no triangles having such $x$.

Is that right?

Added:

My argument is false here. Please refer to the appropriate answer.

Answer 1

Recall that

$$3^2+4^2=5^2 \implies (3\cdot 7)^2+(4\cdot 7)^2=(5\cdot 7)^2$$

and note that

$$(21, 220, 221)$$

is a primitive triple.

Your criterion doesn't works because the remainder of squares $\pmod 4$ are $0,1$ therefore we can't comclude that

$$z^2-y^2\equiv 0 \pmod 4$$

What we need to solve is

$$21^2=441=3^2\cdot 7^2=(z+y)(z-y)$$

that is we need to try with

• $z-y=1 \quad z+y=441\implies (x,y,z)=(21,200,221)$
• $z-y=3 \quad z+y=147\implies (x,y,z)=(21,72,75)$
• $z-y=7 \quad z+y=63\implies (x,y,z)=(21,28,35)$
• $z-y=9 \quad z+y=49\implies (x,y,z)=(21,20,29)$

Answer 2

We have $21=x=k(m^2-n^2),\, y=2kmn,\, z=k(m^2+n^2)$ where $m,n, k \in \Bbb N$ with $\gcd (m,n)=1$ and $m,n$ not both odd.

So $(m^2-n^2,k)\in \{(1,21),(3,7),(7,3),(21,1)\}.$ Now $m^2-n^2=1$ is impossible, so $(m,n,k)\in \{(2,1,7), (4,3,3),(11,10,1),(5,2,1)\},$ giving $$(x,y,z)\in \{ (21,28,35), (21,72, 75),(21,220, 221),(21, 20, 29)\}.$$ We have $m\leq 11$ because if $m\geq 12$ then $x\geq m^2-n^2\geq m^2-(m-1)^2=2m-1\geq 23>21...$ There are 2 solutions $(11,10)$ and $(5,2)$ to $m^2-n^2=21.$

Answer 3

Since $A$ may be any odd number $\ge3$ and we can find one or more triples for any odd leg $A\ge 3$ using a function of $(m,A).$ We know that $A=m^2-n^2\implies n=\sqrt{m^2-A}.$ $$\text{We can let }n=\sqrt{m^2-A}\text{ where }\lceil\sqrt{A}\space\rceil\le m\le \frac{A+1}{2}$$

$\text{Note: }n\in \mathbb{R}\implies \sqrt{A}\lt m\land n<m\implies m\le\frac{a+1}{2}$

$$\text{For A=21}\quad m_{min}=\lceil\sqrt{21}\space\rceil=5\qquad \qquad m_{max}\frac{22}{2}=11$$

Testing for $5\le m\le 11$, we find $\mathbf{integers}$ for $(m,n)=(5,2)\text{ and }(11,10)$

$$f(5,2)=(21,20,29)\qquad f(11,10)=(21,220,221$$ But $21=3*7$

$$\text{For A=3}\quad m_{min}=\lceil\sqrt{3}\space\rceil=2\quad m_{max}=\frac{4}{2}=2\quad \quad 7f(2,1)=7(3,4,5)=21,28,35$$

$$\text{For A=7}\quad m_{min}=\lceil\sqrt{7}\space\rceil=3\quad m_{max}=\frac{8}{2}=4\quad \quad 3f(4,3)=3(7,24,25)=21,72,75$$