## The negative Pell equation and Pythagorean triples.(English)Zbl 0971.11013

Let $$d$$ be a positive integer and $$(A,B,C)$$ be a primitive Pythagorean triple. In this paper, the authors prove that the Pell equation $$x^2-dy^2=-1$$ is solvable in integers $$x$$ and $$y$$ if and only if there exist positive integers $$a,b$$ and an $$(A,B,C)$$ such that $$d=a^2+b^2$$ and $$|aA-bB|=1$$. In this case, $$x=|aB+bA|$$ and $$y=C$$. Similarly, the authors also prove that the number $$\epsilon=(x+y\sqrt d)/2$$, where $$x,y$$ are relatively prime integers and $$x$$ is odd, is the fundamental unit of $$\mathbb{Q}(\sqrt d)$$ with the negative norm if and only if there exist positive integers $$a,b$$ and an $$(A,B,C)$$ such that $$d=a^2+b^2$$ and $$|aA-bB|=2$$, and $$(a,b)=(A,B)=1$$. The method is elementary.

### MSC:

 11D09 Quadratic and bilinear Diophantine equations 11R27 Units and factorization

### Keywords:

Pell equation; Pythagorean triples; fundamental unit
Full Text:

### References:

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