\(p^2+3pq+q^2=m^2\left(m\in N\text{* }\right)\)
\(\Leftrightarrow pq+\left(p+q\right)^2=m^2\)
\(\Leftrightarrow pq=\left(m-q-p\right)\left(m+q+p\right)\)
\(TH1:\left\{{}\begin{matrix}m+p+q=pq\\m-p-q=1\end{matrix}\right.\)
\(\Rightarrow2q+2p-pq+1=0\)
\(\Leftrightarrow\left(p-2\right)\left(q-2\right)=5=1.5\)
\(\Leftrightarrow\left(p;q\right)\in\left\{\left(3;7\right);\left(7;3\right)\right\}\)
\(TH2:\left\{{}\begin{matrix}m+p+q=p\\m-p-q=q\end{matrix}\right.\)
\(\Rightarrow3q+p=0\)
=>không tồn tại p,q thỏa mãn.
\(TH3:\left\{{}\begin{matrix}m+q+p=q\\m-p-q=p\end{matrix}\right.\)
\(\Rightarrow3p+q=0\)
=>không tồn tại p,q thỏa mãn.
Vậy \(\left(p,q\right)\in\left\{\left(3;7\right);\left(7;3\right)\right\}\)
