Polynomial 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...
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 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 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...
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