Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I wanna know, for the equation below, how to:

  • Prove if there is always a natural root $x$ that makes $y$ natural
  • Get the lowest natural $x$ that makes $y$ natural

$$ x^2+8x-y^2=4n-16\quad\forall n\in\mathbb N $$

share|improve this question

1 Answer 1

Note that the equation can be rewritten as $$(x+4)^2 - y^2 = 4n \implies (x+y+4)(x-y+4) = 4n =2n \times 2$$ Hence, one solution is \begin{align} x+y+4 & = 2n\\ x-y+4 & = 2 \end{align} Hence, $$2x+8 = 2n+2 \implies x = n-3$$ $$2y = 2n-2 \implies y = n-1$$ Hence, $(n-3,n-1)$ is always a solution for a given $n$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.