Timeline for What is known about a^2 + b^2 = c^2 + d^2
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 5, 2014 at 10:19 | answer | added | individ | timeline score: 2 | |
| May 9, 2013 at 3:26 | answer | added | Aeryk | timeline score: 5 | |
| May 9, 2013 at 2:13 | comment | added | Noam D. Elkies | $a^2+b^2=c^2+d^2 \Longleftrightarrow a^2-c^2=d^2-b^2 \Longleftrightarrow (a-c)(a+c)=(d-b)(d+b)$. Now the primitive solutions of $rs=tu$ are exactly $(r,s,t,u) = (xx',yy',xy',x'y)$ with $\gcd(x,y)=\gcd(x',y')=1$ (and some positivity condition to avoid duplication with factors of $-1$). If $(a,b,c,d)$ is primitive then $(r,s,t,u)$ is primitive up to a factor of $2$ and satisfies $r\equiv s$ and $t\equiv u \bmod 2$. So just figure out what to do with $x,x',y,y' \bmod 2$ and you're done. | |
| May 9, 2013 at 2:09 | comment | added | Barry Cipra | What qualifies as a "complete" parameterization? This is somewhat unclear (to me, at least) even in the Pythagorean triple case. I know, of course, that $(p^2-q^2,2pq,p^2+q^2)$ is a parameterization for solutions $(a,b,c)$ of $a^2+b^2=c^2$, but is it "complete"? It doesn't include $(4,3,5)$, for example. | |
| May 9, 2013 at 1:16 | history | asked | Favst | CC BY-SA 3.0 |