AM4N :
Yes the Heatsink/Fan I want says it's a 4 pin-connector. Is it possible to buy some sort of adapter? My cpu fan is a 3 pin one
No need for adapter, actually I don't think there is any simple adapter for that.
pin1 & pin2 - +12V and ground, used to power the fan
pin3 - rpm sensor
pin4 - gives PWM control over the rpm
In a 3-pin configuration, the mobo can modify the speed of the fan by adjusting the voltage on pin-1 (+12V line). The less the voltage, the lower the rpms. However, at very low values (like below 5V) the fan will not spin anymore.
In a 4-pin configuration, the mobo can modify the speed of the fan by using PWM signal. The power pin will supply constant +12V voltage, but the 4-th pin will supply a high-frequency 0/1 (logical) signal. This is used to command the rapid power-on and power-off of the fan, depending on the "shape" of this signal. It's like turning the power on and off fast enough so that fan senses it like: a constant power feed, but with less voltage.
Mobo with 4-pin, fan with 3-pin: fan will work at full rpm all the time.
Mobo with 3-pin, fan with 4-pin (your case): fan speed will be controlled by voltage adjustments. No issues, no harm, actually even better than the PWM solution (PWM controlled fans tend to give annoying clicking noise).