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! "An integer is prime iff it appears exactly twice in the infinite times table." Twice twice in the "± times table": Back in 2012-8-30, Subject: zooming in on the reals, I interrupted our regularly scheduled program to bring you the following commercial message:

Apropos nonelectronic calculation, if you have a youngster, hardcopy it this extended times table <http://gosper.org/kellytable.png>. Filling this in should cement intuitions about minus times minus, even times odd, primes, and progressions, e.g..

New! Improved! http://gosper.org/kellytbl.pdf (This message will end in 1.5 KB) When your kid can add and subtract two-digit numbers, and understands the idea of multiplication, s|he might enjoy filling in the rest of this unusual times table. It's easy: just repeatedly add going down or right, and repeatedly subtract going left or up. But don't stop subtracting when you get to zero! What's unusual is that this table contains both negative and positive numbers. You find the multiplier and multiplicand not along the edges, but down in the green rows and columns. E.g., you find (blue) 15 at the intersection of either row containing a green 3 and either column containing a green 5. Likewise at the intersection of a 5 row and 3 column. (For future purposes, all the square numbers are red.) Note that primes can only be green. (1 is NOT a prime!) Filling this in is a useful drill, and it self-checks: You'd better get the same answer whether you reach a box along a column or a row. Farther down the road, the big deal is that this exercise convinces you that negative times negative equals positive, a 7th grade idea. So 15 is also -3 times -5 and -5 times -3. The table also shows how odd times odd = odd (pale pink), and even times anything = even (yellow). And, of course 0 times anything is 0. Exercise: Where is 0 times 0? And when the kid gets it all filled correctly, s|he can show it off at school and maybe drum up additional customers. --rwg Apologies to recipients of a recent version with only a single green row and column. I wrote a pdf in the vain hope that you could paste the InString into Mathematica, but Adobe has other ideas. Also, reading the pdf with Firefox fails intermittently, at least for me. Puzzle: Why does this absolutely dry and crispy snack I am munching list water as an ingredient?