To finish it in 2-3 weeks you basically have to have everything go as planned (which rarely ever happens). Weather, maintenance, etc... Now 4-5 weeks is certainly doable. I did numerous PPL students this summer and finished them in 4-5 weeks. Starting training with the written and ground study already done is a huge help and will make those few weeks of training a lot less painful for you, as at that point you won't be learning everything for the first time but just putting "pen to paper" so to speak.