In the following link we can read (highlighted) about Diophantine equation $A^4 + B^4 = C^4 + D^4$ that it has no "general" parametric solution and yet in next sentence we can read that equation $x^4 + y^4 = u^4 + v^4$ has general (but incomplete) parametric solution.
Diophantine Equation--4th Powers
Can you explain what the author probably meant? If I say general solution I would probably mean complete solution but the author uses "incomplete general solution". What is it?
What of the four following combinations has any sense and which?
- incomplete general parametric solution
- complete general parametric solution
- incomplete non-general parametric solution
- complete non-general parametric solution
PS: Some text should have been highlighted in the link but it does not seem to work. I do not know why.
PS2: There is a bug on your site... in the Hyperlink the character ~ is changed into %7E which spoils scrolling to chosen text and highlighting it.
COMMENT.- A complete general parametric solution is for example given by the very known Pythagorean triples $$x=s^2-t^2\\y=2st\\z=s^2-t^2$$ solving the equation $x^2+y^2=z^2$.
On the other hand the parameterization $$x=m-5\\y=-m+14\\z=3m-30\\w=-3m+29$$ solves partially only the equation $$n=x^3+y^3+z^3+w^3$$ of the representation of a rational integer as a sum of four cubes which is conjecturally analogue to Lagrange's theorem for the squares. This is an example of incomplete general parametric solution because it represents only the integers of the form $18m+8$.