iPhone 4 asks me to log into an Apple address that Apple say doesn't exist I have an Apple ID and a password that works fine on most things but not my iPhone 4. The address is example24@example.com. 

But When I start up my iPhone, iCloud asks me to enter the password for example22@example.com. As far as I know I have never used this email address as an Apple ID. Consequently I have no idea what the supposed password is. 

I have researched your excellent articles about changing apple id addresses but in every case it asks me to log out by entering the old password which I don't know. 

When I try to change the password I get the message that example22@example.com is not an Apple ID.  

In a perfect world all I want to do is completely delete any reference to example22@example.com but I can't do that without knowing what the password for that address is. If anybody can help I would be really grateful.
Start by recovering the password to example22@example.com at https://iforgot.apple.com (and use the guidance provided there in case you don't know the answers to the security questions). Then proceed with decoupling your phone from the old ID.

If you are unable to recover the password to example22@example.com, Apple can remove the old AppleID from your phone. You need to show proof of purchase for this though.