23 still isn't prime (sometimes)

who: J1 and J2
where: walking around the neighborhood
when: after lunch on a saturday

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In response to our earlier discussion that 23 is prime in the integers but not in the rationals, someone on Google Plus (either Curious Cheetah or Paul Hatzer) mentioned that 23 isn't prime in base 9. That led to another good discussion with the kids.

We went through a bunch of bases larger than 3 and discussed whether 23 was prime in that base. A different way of seeing this is that we were looking for primes in the arithmetic series 2b+3:

Base	Prime?
4	Yes
5	Yes
6	No
7	Yes
8	Yes
9	No
10	Yes

They were really excited to feel that they had found a prime-making machine. At this point it seemed that a clear pattern had emerged: all primes except when the base is a multiple of 3. I asked if they had any ideas why and they quickly identified that, when the base is a multiple of three, then 3 will divide 2b+3. We went back to some lower bases for which 23 wouldn't be sensible, but we saw that our prime/non-prime pattern still held:

"Base"	Prime?
1	Yes
2	Yes
3	No
4	Yes
5	Yes
6	No
7	Yes
8	Yes
9	No
10	Yes

Finally, of course, we had to see the bad news: this sequence doesn't hit primes on every base that isn't a multiple of 3 and, in fact, our earlier exploration had stopped just short of the first counter example:

"Base"	Prime?
1	Yes
2	Yes
3	No
4	Yes
5	Yes
6	No
7	Yes
8	Yes
9	No
10	Yes
11	No (boo hoo)
12	No
13	Yes