Answer by Bogdan Grechuk for Polynomial parametrization of the solutions to...
I will prove that, more generally, for any integer parameters $a,b,c,d$, the solution set to any equation in the form$$ax^2 + bx + c = dyz$$is a finite union of polynomial families.Lemma: Let $S...
View ArticleAnswer by Fedor Petrov for Polynomial parametrization of the solutions to...
For $x^2+x+1=yz$ we may factorize LHS in the unique factorization domain $\mathbb{Z}[\omega]$, where $\omega=e^{2\pi i/3}$: $$(x-\omega)(x-\omega^2)=yz.$$Denote by $A$ the greatest common divisor of...
View ArticleAnswer by Tomita for Polynomial parametrization of the solutions to...
This is a partial solution. The equation$$yz=x^2+x+1\tag{1}$$has a parametric solution.Since $x = 1/2 \pm 1/2\sqrt{-3+4yz}$, the expression $-3+4yz$ must be a perfect square. On the other hand,...
View ArticlePolynomial parametrization of the solutions to $yz=x^2+x\pm 1$
If a Diophantine equation has infinitely many integer solutions, how to describe them all? One standard approach is polynomial parametrization. For example, all integer solutions to the...
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