Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there two solutions $m_1,n_1,m_2,n_2$ where $m_i,n_i$ are coprime and $(m_in_i,pq)=1$ and we get $m_i,n_i=O(\max({p^{t-1}},{q^{t-1}}))$ and $m_1n_2-m_2n_1\neq0$?

Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there a solution where $m,n$ are coprime and $(mn,pq)=1$ and we get $m,n=O(\max({p^{t-1}},{q^{t-1}}))$?

Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there a solution where $m,n$ are coprime and $(mn,pq)=1$ and we get $m,n=O(\max({p^{t-1}},{q^{t-1}}))$?

Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations $$m'm^2=1+k_1p^{t-2}q^{t}$$ $$m'n^2=1+k_2p^tq^{t-2}$$ $$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there two solutions $m_1,n_1,m_2,n_2$ where $m_i,n_i$ are coprime and $(m_in_i,pq)=1$ and we get $m_i,n_i=O(\max({p^{t-1}},{q^{t-1}}))$ and $m_1n_2-m_2n_1\neq0$?