57 | 91 | 117 | 119 | 133 | 143 |

147 | 153 | 161 | 171 | 187 | 189 |

207 | 209 | 221 | 231 | 247 | 253 |

273 | 289 | 299 | 323 | 357 | 361 |

391 | 399 | 437 | 441 | 483 | 529 |

57 | 91 | 117 | 119 | 133 | 143 |

147 | 153 | 161 | 171 | 187 | 189 |

207 | 209 | 221 | 231 | 247 | 253 |

273 | 289 | 299 | 323 | 357 | 361 |

391 | 399 | 437 | 441 | 483 | 529 |

My favorite is 119. I still don’t believe it’s not prime.

Yeah, 91 and 119 always get me, too. I call them “Zucker pseudoprimes” — which is to say, numbers that Zucker thinks are prime but actually aren’t. Of course, as soon as I know that number is on that list, it isn’t on that list any more.

I think a little bit of mental arithmetic habits will kill off a lot of these, though. For instance, I know my 11s tables far enough to get 143. I reflexively sum up digits enough to get all the ones here that are divisible by 3. I know my squares well enough to catch 289, 361, 441, and 529. And I recognize difference of squares fast enough to get 221, 391, 399. Let’s see, what’s left for me now?

91, 119, 133, 161, 181, 187, 209, 231, 247, 253, 299, 323, 437.

A bunch of these are pretty quick if I check my 7s. That takes care of 91, 119, 133, 161 at least. 181 I’ll save for later. Knowing the 11s rule gets me 187, 209, 231, 253. So now I’m down to 181, 247, 299, 323, 437.

Wait a minute, 181 really is prime!

247 is 256 – 9 I guess, so there’s difference of squares, but that’s not so easy to see.

299 just looks nasty. I don’t think seeing the 324-25 difference of squares is all that realistic for me, but maybe I need to get a little more aggressive with my squares.

323 is 324-1, should have noticed that earlier with the difference of squares.

437 is 441 – 4.

Seems like the lesson is that I should know my squares REALLY WELL. That would save me from 91 for sure, 100 – 9. And 119? Hm. 144-25 seems tough to see.

I should shortcut this a bit by thinking “3 mod 4, so add an odd square to get an even square” and then I need to just check 120, 128, and 144 for this one, and similarly for 299 I check 300, 308, 324. And for the things that are 1 mod 4, I try adding the even squares.

Woops! Thanks — I meant 171, not 181. I’ve corrected the table. I appreciate your analysis—it actually backs up my affection for 119, which was totally offhand! (That is, as long as I refuse to learn my 7′s….)

Hmm, why isn’t 39 on here? Since multiples of 3 are, and no one learns times tables past 12. I love the number 1001, because …

Maybe when all the digits are divisible by 3, it’s clear enough that the number is divisible by 3? To me, 39 seems a lot more obviously divisible by 3 than 57 or 483.

@Sue – What Joshua said. Also 39 is on my list of sight products – I think students will be well served if they can instantly recognize 26, 39 and 52 as multiples of 13. The rest of the 13′s haven’t shown up much in my experience.