Bill Gosper wrote:

ES>

According to a friend who volunteers in the Santa Cruz CA Public Schools, the official view is that 1 is a prime, because its only divisors are 1 and itself. However, a teacher did mention that in more advanced mathematics, 1 is not a prime.

 -- Gene

We need to displace that definition!

Well, the official definition of `prime' is `for all a,b in R, we have p|ab <==> (p|a or p|b)'. I think that could possibly be de- algebraicised to make it compatible with 4th-graders (whoever they are; I don't know how American grades translate into actual ages). Of course, you probably want to use the definition of `irreducible' instead (they coincide in the integers, or more generally any PID), in which case it is `we cannot decompose p into a product of two factors, neither of which is a unit'. So: `A positive integer is prime if it cannot be decomposed into a product of two factors, both of which are greater than 1.' Or equivalently: `A positive integer is prime if it cannot be decomposed into a product of two strictly smaller factors.'

"An integer is prime iff it appears exactly twice in the infinite times table." Twice twice in the "± times table":

It might be difficult to make the conceptual leap from that into a general intdom, though, since there is no nice generalisation. (I suppose if there are finitely many units, as is usually the case, you can say `appears exactly 2n times in the times table'.) Anyway, considering how many times things appear in the Cayley table of the monoid of positive integers under multiplication is quite nice. A _Klein number_ (square of a prime) is something that appears exactly thrice, for example. (The reason for the name is because Felix Klein was born on 1849-04-25, and 4, 25 and 1849 are all squares of primes.)

Puzzle: Why does this absolutely dry and crispy snack I am munching list water as an ingredient?

Being edible, I imagine that it is composed of biological cells (plant? animal? fungus?), and the cytoplasm is full of water. As Martin Chan astutely remarked, `Oh, so ROM bars are 40% rum in the same way that a tomato is 90% water?'. (The rum is absorbed into a brown paste permeating the chocolate bar.) http://lurnq.com/lesson/ROM-The-American-Takeover-Case-Study/ They're also tasty and conducive to solving mathematical problems. Indeed, I can't think of another chocolate bar to which I am equally indebted. I suppose I should acquire one next time I contemplate this question: "Let alpha be an ordinal. Does there exist a function from the set of limit ordinals below alpha to the power-set of alpha such that the following properties hold? 1. f(beta) is a subset of beta for all beta; 2. If beta and gamma are distinct, f(beta) is not a subset of f(gamma)." (The second condition is essentially saying `the image of f forms an antichain under inclusion'.) The answer is certainly `yes' up to the first weakly-inaccessible cardinal, so there are models of ZFC in which the answer is `yes' for all ordinals. However, I don't know whether it is provably true for all ordinals in ZFC alone. Sincerely, Adam P. Goucher http://cp4space.wordpress.com