I have a laptop with these specifications :
Intel i5 5200u
4gb ram
Intel hd 5500
Windows 8.1 64bit
500gb storage
Will this laptop run UE4 ?
Test it out
you may be able to run it but it wont run well and i dont think you will be able to get the latest versions to even open. your computer doesnt meet the recommended hardware spec since you have less ram, laptops arent supported, and you dont have a dedicated graphics card. the graphics card will be one of the most limiting factors here. without a graphics card you force the graphics rendering to be done on the cpu which takes away performance from what it should be working on which will greatly slow things down. the next limiting factor is the ram, without more ram you wont be able to work with many things in the editor such as having a few blueprints open and having many things in your scene or decent rez textures. also video cards usually have their own built in memory which will also need to be stored in ram now since you dont have a gpu.
overall its not recommended to use that pc for unreal. but like others have said you can always download an older version and try running it to see what happens. if you do try an older version id say to go with a version older than 4.17. if it doesnt work then you can always try udk or some other engine.