I had an iPhone that was stuck on a recovery loop. That means it would be in restore/recovery mode, where it would refuse to boot into a usable state, but instead frustratingly display a prompt to connect to iTunes every time it was turned on.

Every blog and forum post recommended a myriad of different fixes: iREB, iRecovery, iPwnage came up most frequently.

None of them worked. Oddly, the simplest fix was to download the latest firmware IPSW file from Apple. It's safe, it's from Apple and it does not jailbreak your phone. (It's what iTunes would have downloaded in the background, but you're getting the file directly now because your copy is probably corrupted, leading to the 1604 error.)

1. Download the right IPSW for your phone. Here's a site which lists a bunch of them.
2. Connect your phone to iTunes and say yes to all the prompts which say you need to restore your phone.
3. When you get to press "restore", hold down alt+restore for Macs (Windows users: hold down shift).
4. Select the IPSW you've downloaded.
5. Pray!

Good luck!