I absolutely promise not to post on this topic any more. But having
said something stupid I have to take it back, which requires posting,
& trying to avoid saying something stupid . . . I think this is
called karma.
What I neatly demonstrated had nothing much to do with the problem,
because I was counting prime factors.
For a proof that the Ray Theorem is true, I refer anyone interested
to a friend & colleague who's a mathematician and whom I should have
asked before I posted anything, and who wrote as shown below.
Charles, shutting up
================= snip =============
Anyway, here’s an argument. Consider the positive integer n > 1.
There are two cases to
consider.
1. Suppose √n is not an integer. (Then, of course, √n is not even
rational but that is
another story.) Now n has at least one factor that is smaller than
√n, namely, 1. And,
clearly, for each distinct factor of n smaller than √n, there is a
distinct factor of n larger
than √n such that their product is n. Thus, if n is not a perfect
square, n must have an
even number of distinct factors.
2. Suppose √n is an integer, i.e., n is a perfect square. Then
arguing as above, for each
distinct factor of n smaller than √n, there is a distinct factor of n
larger than √n such
that their product is n. Therefore, other than the factor √n itself,
n has an even number
of distinct factors. But, √n, in this case, is also a factor on n.
Therefore, if n is a perfect
square, n must have an odd number of distinct factors.
This proves that a positive integer n has an odd number of distinct
factors if and only if
n is odd.
Now, since 1412 = 19881 and 1422 = 20164, there must be exactly 141
lights that are left
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