What Diophantine equations employ Pell equations in their solutions? A well-known example is the case of Pythagorean triples where the legs differ by 1, like,
$$20^2+21^2 = 29^2$$
These are completely parameterized as,
$$\big(\tfrac{x-1}{2}\big)^2+\big(\tfrac{x+1}{2}\big)^2=y^2$$
where $x^2-2y^2 =-1$. These $x,y$ also imply,
$$x^4+(y^2-1)^2 = y^4$$ $$x^4+(2y)^2 = (2y^2)^2+1$$ $$1^3+2^3+3^3+\dots+(x^2)^3 = (xy)^4$$ $$1^3+3^3+5^3+\dots+(2y-1)^3 = (xy)^2$$
Other equations that are to be solved in the integers and use Pell equations (at the least as a partial solution) are in the list below. (It is to be understood that $a,b,c,d$ are constants, while $x_i,\,y_i,\,z_i$ are unknowns.)
I. Simultaneous equations
$x^2+y^2-1,\;x^2-y^2-1,\;\text{both squares}$
$xy+1,\;xz+1,\;yz+1,\;\text{all squares}$
$xy-1,\;xz-1,\;yz-1,\;\text{all squares}$
II. Quadratics
$ax^2+bx+c = dy^2$
$ax^2+bxy+cy^2 = d$
$ax^2+bxy+cy^2 = dz^2+e$
$x^2-x^2y^2+y^2=1$
$x_1^2+x_2^2\pm1 = y^2 $
$x_1^2+x_2^2+\dots+x_m^2\pm k = y_1^2+y_2^2+\dots+y_n^2$
III. Higher Powers
$x^2+y^3 = z^4$
$x^4+y^3 = z^2$
$x^3+y^3+z^3+x+y+z = 0$
$x_1^3+x_2^3+x_3^3 = 1$
$x_1^5+x_2^5+\dots+x_7^5 = 1$
$x_1^7+x_2^7+\dots+x_9^7 = 1$
$x_1^4+x_2^4 = y^2+1$
$x_1^4+x_2^4+x_3^4 = y_1^4+y_2^4+1$
$x_1^6+x_2^6+x_3^6+x_4^6 = y_1^6+y_2^6+y_3^6+1$
Q: What other examples are not covered in the list above?
