Your Lightroom 4 license serial number won't work.
Your Lightroom 5 license serial number will work if you download the proper version. If you bought the standalone version (also called "perpetual license"), you need to download it from here: Product updates
If you bought the Creative Cloud version of Lightroom 5, you don't need a serial number, but you do need to sign in.
Because it's an LR 4 license. You will have to purchase an LR 5 license. The upgrade price is $80 in the US.