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For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?

Is $gcd(a,b)=gcd(c,d)=1$ possible?

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?

Is $gcd(a,b)=1$ possible?

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?

Is $gcd(a,b)=gcd(c,d)=1$ possible?

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$$$0<|a|^2,|b|^2,|c|^2,|d|^2<|x|,|y|$$ 

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?

For a given $t\geq4$, does the following system of equations have a solution over the integers? $$ax^2+by^2=2^{2^t-t}$$$$cx^2+dy^2=1$$ $$0<|ta|^2,|tb|^2,|tc|^2,|td|^2<|x|,|y|$$

If so, how to parametrize the solutions and find them?

For a given $x,y:|x|,|y|<B$, how many such $a,b,c,d$ are there?