(These also have an associated primality conjecture, that is prime whenever is, although it's much quicker to see that this is false. And the converse, that is prime whenever is, is actually true! Since , this even lets you be perfectly ambiguous about whether to accept as prime.)

]]>But I like how this bootstraps the beginning of the Fermat numbers. I say ‘I want a Fermat number!’. You say ‘Well, just multiply together the ones that you have, then add 2.’ (which doesn't work if I obtain them out of order, but never mind that for now). I reply ‘But I don't have *any* Fermat numbers yet!’, but you reply ‘Don't worry, just do what I said: multiply all none of them together to get 1, then add 2 to get 3.’ and now I've got my first Fermat number out of nothing.

(As the original version of your comment noted, 1 could itself be considered a non-prime Fermat number, since , but that breaks both puzzles, so let's forbid it.)

]]>But I like how this bootstraps the beginning of the Fermat numbers. I say ‘I want a Fermat number!’. You say ‘Well, just multiply together the ones that you have, then add 2.’ (which doesn't work if I obtain them out of order, but never mind that for now). I reply ‘But I don't have *any* Fermat numbers yet!’, but you reply ‘Don't worry, just do what I said: multiply all none of them together to get 1, then add 2 to get 3.’ and now I've got my first Fermat number out of nothing.

(As the original version of your comment noted, 1 could itself be considered a non-prime Fermat number, since , but that breaks both puzzles, so let's forbid it.)

]]>