The original doomsday conjecture was that for any prime p and positive integer s there are only a finite number of permanent cycles in
Mahowald (1977) found an infinite number of permanent cycles for p = s = 2, disproving the conjecture. Minami's new doomsday conjecture is a weaker form stating (in the case p = 2) that there are no nontrivial permanent cycles in the image of (Sq0)n for n sufficiently large depending on s.
Milgram, R. James (1971), "Problems presented to the 1970 AMS symposium on algebraic topology", in Liulevicus, Arunas (ed.), Algebraic Topology, Proc. Symp. Pure Math, vol. 22, pp. 187–201
Minami, Norihiko (1998), "On the Kervaire invariant problem", in Mahowald, Mark E.; Priddy, Stewart (eds.), Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), Contemp. Math., vol. 220, Providence, R.I.: Amer. Math. Soc.,
ISBN978-0-8218-0805-4,
MR1642897