How does one analytically determine the integer values of a rational function $f(x)$$=$$\frac{40-8x}{8x+2}mod1$ where $x$ is an element of the rationals? I just gave the function listed as an example, I would like to know the generally preferred methods (if they exist) of analytically determining integer values of rational functions.
Asked
Active
Viewed 236 times
0
-
Did you really mean $\mod{1}$? – Ethan Hunt Mar 29 '16 at 03:24
-
@EthanHunt Yes, as $f(x)$ doesn't take on only integer values for rational (or even integer) $x$. – user3108815 Mar 29 '16 at 03:33
-
1This is not "modular-forms", nor "modular-arithmetic". As I understand it, you simply want to find when $(40-8x)/(8x+2)$ is an integer. – Robert Israel Mar 29 '16 at 03:51
-
@RobertIsrael edited tags to reflect. How does one obtain the integer solutions though analytically? – user3108815 Mar 29 '16 at 03:56
1 Answers
1
Multiply the equation $\frac{40 - 8 x}{8 x + 2} = y$ by the denominator and you have a linear equation in $x$: solve to get
$$ x = \dfrac{20-y}{4+4y} $$ Note that if $y$ is an integer (other than $-1$), $x$ is a rational number.
Robert Israel
- 448,999