I guess, though I am not sure, the case of Albert Wormstein falls in your third category:

Professional mathematicians writing mathematics under both their real name and a pseudonym.

This paper: "Polyominoes of order 3 do not exist" (Journal of Combinatorial Theory, Series A, Volume 61, Issue 1, September 1992, Pages 130–136) has been written by I. N. Stewart and A. Wormstein.

Here is the story behind the paper as told by Ian Stewart himself.

The link has the correct story. Albert Wormstein first appeared in one of my articles for Pour La Science / Scientific American, which was used as a chapter in the cited book. While I was writing that article it suddenly seemed clear that there ought to be a way to prove the conjecture about order 3 polyominoes. It felt as though Albert was tapping me on the shoulder and saying 'come on, we can do this.' It quickly turned out he was right. So I decided to give him credit as a co-author. The journal either spotted the joke and went along with it, or they assumed Albert was a PhD student. At any rate, they published it with him as co-author.

I guess, though I am not sure, the case of Albert Wormstein falls in your third category:

Professional mathematicians writing mathematics under both their real name and a pseudonym.

This paper: "Polyominoes of order 3 do not exist" (Journal of Combinatorial Theory, Series A, Volume 61, Issue 1, September 1992, Pages 130–136) has been written by I. N. Stewart and A. Wormstein.

Here is the story behind the paper as told by Ian Stewart himself.

The link has the correct story. Albert Wormstein first appeared in one of my articles for Pour La Science / Scientific American, which was used as a chapter in the cited book. While I was writing that article it suddenly seemed clear that there ought to be a way to prove the conjecture about order 3 polyominoes. It felt as though Albert was tapping me on the shoulder and saying 'come on, we can do this.' It quickly turned out he was right. So I decided to give him credit as a co-author. The journal either spotted the joke and went along with it, or they assumed Albert was a PhD student. At any rate, they published it with him as co-author.

I guess, though I am not sure, the case of Albert Wormstein falls in your third category:

Professional mathematicians writing mathematics under both their real name and a pseudonym.

This paper: "Polyominoes of order 3 do not exist" (Journal of Combinatorial Theory, Series A, Volume 61, Issue 1, September 1992, Pages 130–136) has been written by I. N. Stewart and A. Wormstein.

Here is the story behind the paper as told by Ian Stewart himself.

The link has the correct story. Albert Wormstein first appeared in one of my articles for Pour La Science / Scientific American, which was used as a chapter in the cited book. While I was writing that article it suddenly seemed clear that there ought to be a way to prove the conjecture about order 3 polyominoes. It felt as though Albert was tapping me on the shoulder and saying 'come on, we can do this.' It quickly turned out he was right. So I decided to give him credit as a co-author. The journal either spotted the joke and went along with it, or they assumed Albert was a PhD student. At any rate, they published it with him as co-author.

I guess, though I am not sure, the case of Albert Wormstein falls in your third category:

Professional mathematicians writing mathematics under both their real name and a pseudonym.

This paper: "Polyominoes of order 3 do not exist" (Journal of Combinatorial Theory, Series A, Volume 61, Issue 1, September 1992, Pages 130–136) has been written by I. N. Stewart and A. Wormstein.

Here is the story behind the paper as told by Ian Stewart himself.

The link has the correct story. Albert Wormstein first appeared in one of my articles for Pour La Science / Scientific American, which was used as a chapter in the cited book. While I was writing that article it suddenly seemed clear that there ought to be a way to prove the conjecture about order 3 polyominoes. It felt as though Albert was tapping me on the shoulder and saying 'come on, we can do this.' It quickly turned out he was right. So I decided to give him credit as a co-author. The journal either spotted the joke and went along with it, or they assumed Albert was a PhD student. At any rate, they published it with him as co-author.