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.

Finding the positive integer solutions $(x,y)$ of the equation $x^2+3=y(x+2)$

Source: Art of Problem Solving Vol. 2

Any help would be appreciated.

share|improve this question
1  
Considering what $gcd(x+2,x^2 + 3)$ could be would be a help. –  Geoff Robinson Dec 31 '11 at 18:02
1  
What have you tried? –  lhf Dec 31 '11 at 18:53
add comment

2 Answers

up vote 7 down vote accepted

Divide the polynomial $x^2+3$ by the polynomial $x+2$. We get $$\frac{x^2+3}{x+2}=x-2+\frac{7}{x+2}.$$ For an integer $x$, the value of $x-2+\frac{7}{x+2}$ is an integer if and only if $x+2$ divides $7$. The only positive integer $x$ for which this is true is given by $x=5$.

Comment: If we are interested in integer solutions, not necessarily positive, note that $x-2+\frac{7}{x+2}$ is an integer also at $x=-1$, $x=-3$, and $x=-9$.

share|improve this answer
add comment

HINT $\ $ Factor Theorem $\rm\:\Rightarrow\ x+2\ |\ f(x)\ \iff\ x+2\ |\ f(-2)\:,\ $ for all polynomials $\rm\ f(x)\in \mathbb Z[x]$

So, in particular, for $\rm\:x\in \mathbb N\:,\ \ x+2\ |\ x^2+3\ \iff\ x+2\ |\ 7\ \iff\ x\ =\ \ldots$

share|improve this answer
add comment

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.