About 3 years ago I discussed following two Diophantine equations of degree 2:

- A special case of negative Pell’s equation (determining solvability conditions for a special degree 2 equations in 2 variables)
- A slight modification of Pythagorean equation (finding explicit solutions for a particular degree 2 equations in 3 variables, using method used for finding Pythagorean triplets)

In this post, we will see a slight generalization of the result involving Pythagorean triplets. Unlike Pythagoras equation, , we will work with a little bit more general equation, namely: , where . For proofs, one can refer to section 5.5 of Niven-Zuckerman-Montgomery’s *An introduction to the theory of numbers*.

Theorem:Let be non-zero integers such that the product is square free. Then have a non-trivial solution in integers if and only if do not have same sign, and that are quadratic residues modulo respectively.

In fact, this result helps us determine the existence of a non-trivial solution of any degree 2 homogeneous equation in three variables, due to the following lemma:

Lemma:There exists a sequence of changes of variables (linear transformations) so that can be written as an equation of the form with .

Now let’s consider the example. Let , and we want to determine whether this has a non-trivial solution. Firstly, we will do change of variables:

where , and . Thus

where , and . Thus

where , and . Now we apply the theorem to . Since all the coefficients are prime numbers, we can use quadratic reciprocity to conclude that the given equation has non-trivial solution (only non trivial thing to note that is quadratic residue mod -3, is same as is quadratic residue mod 3).

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