# Thread: Polynomial with integer solutions.

1. ## Polynomial with integer solutions.

Am I doing it correctly? What would the next step be?

2. If you let $\displaystyle y = x + k$, then k must be an integer and x must be an integer solution to $\displaystyle x^2 + kx - 12k = 0$. Hence, the discriminant of this polynomial, which is $\displaystyle k^2 + 48k = k(k + 48)$, must be a square. Finally, let $\displaystyle k = n - 24$. Then $\displaystyle (n - 24)(n + 24) = n^2 - 24^2$ must be a square. Now use your knowledge of pythagorean triples and multiples of them to find solutions:

$\displaystyle (3, 4, 5) \to (24, 32, 40); (18, 24, 30)$
$\displaystyle (5, 12, 13) \to (10, 24, 26)$
$\displaystyle (7, 24, 25)$
$\displaystyle (8, 15, 17) \to (24, 45, 51)$
$\displaystyle (9, 40, 41)$
$\displaystyle (11, 60, 61)$
$\displaystyle (12, 35, 37) \to (24, 70, 74)$
$\displaystyle (24, 143, 145)$

Now we need hypotenuses $\displaystyle n > 24$. From these triples we get the solutions $\displaystyle 25, 26, 30, 40, 51, 74, 145$.

This yields the following solutions for k:
$\displaystyle 1, 2, 6, 16, 27, 50, 121$

That gives you seven quadratic equations to solve for x, some of which may not yield integer solutions.

3. Hence, the discriminant of this polynomial must be a square.
Why?

Edit: Quadratic formula? Since we want integer solutions, the discriminant must be a perfect square?

Now use your knowledge of pythagorean triples and multiples of them to find solutions:
Sorry, but solutions to what? (n-24)(n+24)? or $\displaystyle n^2 = 24^2$? In either case, where do Pythagorean triples come into play?

Edit: Pythagorean Theorem. $\displaystyle n=24, (n)^2 - 24^2 = k^2 \implies k^2 + 24^2 = n^2.$ Is that about right?

Now we need n > 24.
I'm assuming the answer to this follows from question (2) above.

Edit: $\displaystyle k = n - 24, k > 0$, so $\displaystyle n > 24$. Although, shouldn't this go without saying since $\displaystyle n = 24, n^2 = 24^2 + k^2 \implies k = 0$, which doesn't make sense?

Thanks so much for your help.

4. Sorry, that was a hopelessly convoluted method. You are looking for any value of n such that $\displaystyle n^2 - 24^2 = a^2$ for some integer a.

I am editing my post to take out values of n that don't work. Where the Pythagorean triples come in: n is the hypotenuse of a right triangle, with one side equal to 24, and the other side equal to a.

It is much easier to observe that x must be an integer from 1 to 11, and try each of those values and determine whether y ends up being an integer.

5. One quick question: Is there another way to solve this without the knowledge of pythagorean triples that I don't have?

Thanks

6. Hello !

You want to solve $\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{12}$ for $\displaystyle x$ and $\displaystyle y$. This is how I would have done :

$\displaystyle \frac{1}{x} = \frac{1}{12} - \frac{1}{y}$

$\displaystyle \frac{1}{x} = \frac{y}{12y} - \frac{12}{12y}$

$\displaystyle \frac{1}{x} = \frac{y - 12}{12y}$

$\displaystyle x = \frac{12y}{y - 12}$

All you have left is to determine for which values of $\displaystyle y$ do we have an integer $\displaystyle x$, that is, when $\displaystyle (y - 12) | 12y$

Does it make sense ?

7. Originally Posted by Bacterius
Hello !

You want to solve $\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{12}$ for $\displaystyle x$ and $\displaystyle y$. This is how I would have done :

$\displaystyle \frac{1}{x} = \frac{1}{12} - \frac{1}{y}$

$\displaystyle \frac{1}{x} = \frac{y}{12y} - \frac{12}{12y}$

$\displaystyle \frac{1}{x} = \frac{y - 12}{12y}$

$\displaystyle x = \frac{12y}{y - 12}$

All you have left is to determine for which values of $\displaystyle y$ do we have an integer $\displaystyle x$, that is, when $\displaystyle (y - 12) | 12y$

Does it make sense ?
Yes, but you copied the problem incorrectly.

8. Oh, yeah. Well it doesn't actually change much, just replace the signs