not A + J = T but Tytos's wife + some Targ = Tywin the bastard

[The Others Take Ya]

I'm not sold on the A + J = T theory, because I don't think it accounts for the similarities between Tyrion and Tywin, plus the Jamie and Cercei hints. However, my thought is that it's more plausible that Tywin Lannister is actually a Targaryen bastard, making the kids, Jaime, Cercei and Tyrion 1/4 Targ. That would account for all of them showing different hints of Targaryen tendencies in the text. And they would still be half Lannister from their mother Joanna's side.

Tytos Lannister is described as weak willed and seems very different then Tywin in most things, and could be someone who is likely his wife would cheat on.

Look at Genna Lannister on the wiki! No one wants to ask if her kids are actually Freys! lol Just like mom perhaps?

There are some unknowns around Egg's children, and his brothers and sisters children in the Targ family tree and Aerion the crazy one from D&E had a one year old son that was passed over around the time of succession. These are around the generation and time period of Tywin's birth.

It also seems likely that Aerys would want a rumored Targaryen bastard as his hand of the king, thinking that, like Bloodraven, it would be a wise choice to keep the duties of ruling in the family. Whether Tywin knew, suspected or suppressed his own bastardy during his life, Aerys may have known through his side of the family.

Narratively it would be kinda awesome if Tywin who was so concerned about the family name, was actually not a Lannister the whole time!

History would repeat itself if Tyrion was a kinslayer, bastard, and hand of the king like Bloodraven (who killed his brother Bittersteel).

I'm not saying I believe this one way or the other BTW, I just think its more likely. I just searched and searched for a thread on this possibility and couldn't find it. I can't be the first one to think of this, but I thought it deserved its own thread.

edit: spelling mistake, clarity