Associative Hooks

The very first thing to notice is that remembering products is all about association. When I say I've learned the product 893 = 19×47, it means that when I see 893, I immediately think of 19 and 47 … in other words, that I've created associations from 893 to 19 and 47. To create the associations, however, first of all I needed to have something at the other end, something I could hook the associations to. In other words, I needed to have, or develop, good associative hooks, good mental representations of the various primes.

• I already had good representations of 2, 3, and 5 … so much so that sometimes I even think I understand their intrinsic nature. Unfortunately, those primes aren't relevant here, because the whole idea of products is that factors of 2, 3, and 5 are handled differently.
• I had some cultural associations for 7, 11, and 13: the first two are lucky, the third, unlucky. Also I sometimes used divisibility tests for the first two.
• I had a literary association for 17, and was later given one for 19: in the world of Dragaera, there are seventeen noble houses; and in the world of the Lensmen, you know your calculations are correct when they match to the proverbial nineteen decimal places.

  Also, there's a cereal named Product 19, ha ha.

• To begin with, I had hardly any associations for 23, 29, and 31, but then I developed a few as I worked. As it happens, most products containing 23 are pleasing and easy to remember, …

  161 = 7×23

  253 = 11×23

  299 = 13×23

  … but many products containing 29 and 31 are ugly and difficult. I ended up thinking of the three as being like Cinderella and her evil stepsisters.

  Speaking of evil, 19 is sort of like an evil twin of 17, because if you're learning the products in order, every product for 17 is soon followed by a confusingly similar product for 19.

• 37 is distinctive because it goes evenly into 999.
• That leaves 41 as the first nondescript prime. That's a joke, or maybe a paradox … the phrase “the first nondescript prime” is pretty descriptive, isn't it? I know I've seen that paradox somewhere else, expressed as a statement about the first uninteresting number, but I can't figure out where. The best I can do is point out that it's related to Berry's paradox, which is mentioned in The Unexpected Hanging, and is about the least integer not nameable in fewer than nineteen syllables.
• Later, when I started thinking about four-digit numbers, I realized that 43, 53, and 59 were special because of how they participated in those three combined divisibility tests. That leaves 41 and 47 as gaps, and 61 as the first nondescript prime.

I did eventually develop associative hooks for other, larger primes, but the hooks are mostly weak, and don't have much content … they're basically defined by the products that are attached to them. Here are the few that have other associations.

67	the start of the decimal expansion of 2/3, approximately
97	see below
10_	that group of four primes
113	that prime below that large gap
127	that prime above that large gap; also a Mersenne prime
137	the reciprocal of the fine structure constant, approximately
139	the largest prime used in three-digit products
19_	that other group of four primes
211	the largest prime used in the products up to 1500

One nice thing about three-digit products (and about the products up to 1500) is that the smallest prime factor is never larger than 31 (37), so at least one factor is guaranteed to be a strong hook. That's not ideal, but it's significantly better than nothing. For four-digit products, the smallest prime factor is never larger than 97.