Do that long enough and you find patterns, like 7 * 11 * 13 = 1001. If you ever end up calculating n / 11, it's approximately the same as n * 7 * 13 / 1000. E.g., 3 / 11 ~= .273. Or take 27 * 37 = 999. Now n / 37 ~= n * 27 and shuffle the decimals. 7 / 37 ~= 7 * 27 / 1000.
And that's how I ended up reasonably good at mental arithmetic, and memorizing a frankly unnecessary number of squares, and being able to factor lots of numbers at a glance (or recognize that they're prime). I was awfully bored for an awfully long time.
Imagine being an ancient Babylonian or Greek with nothing to do but wait for your grapes to grow. No wonder they came up with lots of great stuff.
I agree, and I worry that the absence of boredom-time, especially for kids and adolescents, will turn out to be a bad thing.
I've never met a kid or adolescent who shares this concern. :)
These primes have to be 11,13,17,19 + 30k, due to division by 3.
5 is half of 10, so we easily rule out exactly the number 10x+5.
49 = 7²,
77 is a multiple of 7, of course.
98 = 2 • 7²
In order to avoid a multiple of 11 interfering, we need the gap preceding 11•10
99 is (10+1)(10-1) = 10²-1 and 100 is 10²
, so that clears out space for primes after 100.
Also, You might enjoy "Paterson Primes" https://m.youtube.com/watch?v=jhObLT1Lrfo
so I'll tell you how I memorized the primes between 2 and 127.
to begin: all primes less than ten: 2,3,5,7. I seem to have learned these by rote memorization. but then, the 'fun coincidences' begin.
because 5 is a multiple of "ten", all primes after 10 can only end in 1,3,7,9. this is a key somehow.
this gives the first fun coincidence: that 11 and 101 are both prime.
after 19 comes 23. similarly (and this is the part where every self-respecting numberphiliac waves their hands a little, in excitement I hope), 113 ("one one three") is the next prime after "one oh nine".
notice that 23,29,31,37 are as much prime as 30+{23,29,31,37} = 53,59,61,67
this covers almost all of them. but we're missing primes in the 40s, and the 70s. as well as primes in the 80s and 90s (only 3 primes: 83, 89, and 97. I have no tricks to remember these 3. only rote repetition)
the forties and seventies, are a very compact: both 1,3. but only 47, and only 79; because, well, what fun! seven squared and seven eleven are just so simple, like low-hanging fruit in the garden of prime-number coincidences
Conversely, to me, 9733 “looks prime”. I couldn’t explain that if I had to. It just does.
I like that there’s enough prime real estate for us all to see it differently.
doing it in different bases helps to think about the numbers without their decimal representation. which I guess is the point, and might be helping me imagine patterns that may or may not be there. if I learned anything from doing this is that there is no pattern, it just seems as if there is one but it's never really there.
I like to think that the "patterns" expire, they have only so many uses in them. often just one use which breaks the point of being a 'pattern', but that's primes; is all am saying.
the problem is that I don't know why do this.
nonetheless I can report that I've memorized 3 instances of two consecutive twin primes
11,13,17,19, then 101,103,107,109 (which just raises questions that I can only aspire to ask, nevermined answering, about the what, why, and how of decimal system),
and then 191,193,197,199. the next prime is 211. but the cool thing is how 210 = 2*3*5*7, which are all primes before the first double twin prime