2323
Imagine, if you will, a number line. This number line begins at zero, at extends off into the distance, as number lines do. Somewhere on that line, is the integer 2323. This is an inclusive number line-it includes all types of numbers: rationals, irrationals, integers, you name it.
This number under consideration, 2323, is interesting, because it repeats 23 twice (the so-called “Jordan Prime”, or “Lebron Prime”, depending on how old you are). We can make this number by beginning at zero, and starting to adding ones. This process, of adding +1, +1, +1, over time, will get us “down the line” to 2323, but it is slow, and tedious. Arithmetic is faster and more efficient counting, but the slow process of building on by one, although scenic, if you are into linear trips along straight lines, passing by an infinite number of other numbers along the way.
You will pass such luminary numbers as 1/2, e, pi, 20, 50, 100, sqrt10002, 1000, and many others, along the way. Travelling in one dimension isn’t that interesting, though- you look left, and you look right, or at least you try to. You can’t. Life in one dimension is very one dimensional- you can look forward, and you can look back, but that’s about it. Maybe not even that-you would have no thickness, after all.
Multiplication can get us there much faster. The trip to 2323 will take much less time. If it is prime, we just do it all at once: 1 x 2323 and we get there, in much less time, than it would take to add all those ones. Consider this a bit like taking a giant hop with giant boots all the way to where you want to go (but not really, as you are in one dimension).
If 2323 is not prime (and it seems like it could be-it ends in “3”, and it consists of a prime repeated twice, which could mean everything, or nothing at all, as numbers are very mysterious, and have personalities of their own.)
It seems like 2323 should be, or could be prime, but is it?
If 2323 has factors other than one and itself, we could jump to the first of these prime factors, say, 3, if it is one, and then scale up the next prime factor. If that is 11, we would scale up 3 11 times. It wouldn’t take too many hops to to get there.
Imagine now we define multiplication over two dimensions. Some rectangle, a x b, could give us 2323.
There exists a long skinny rectangle that is 1 unit by 2323 units. It is the “long skinny cigar” rectangle, like shown left.
You could travel around the outside of this rectangle, if you like, but it would take a really long time to travel 2323 x 2 + 2 units.
There might be other possible rectangles, that give us a shorter walk, however, but how are we find them?
50 times 50 is 2500. That would be the case of rectangle we call a square. Our “squarest” possible rectangle, if it exists, is smaller than 50 x 50. So we need to check all prime factors up to, say, 43 x 43. (Just a guess)
Walking around the outside of squares is often faster than walking around the outside of rectangles. If you are tired, you hope for a more squarish, less cigarish rectangle to exist.
The Sieve of Eratosthenes always helps to find these possible rectangles, but is still too much when numbers get too big (just ask any regular computer that is not a quantum computer).
2 is out as a factor. There is no rectangle with 2 width.
3 is out as a factor. I inspect the digits and see that 2+3+2+3=10, so this number is not divisible by 3.
5 is out, as this number does not end in 5 or 0.
7? 7 x 300 is 2100. We need 223 more. 70 x 3 is 210. We are left with 13, because we land at 2310. Not divisible by 7.
11? I know no tricks for that, but let’s try… 11 x 100 is 1100. 200 x 11 is 2200. We are short 123 from our target. 11 x 11 is 121. Not divisible by 11.
My hopes are now up that this number is prime.
13? 13 x 100 is 1300. 13 x 50 is 650. We are at 1950. 13 x 25 is 325. We are at 2275. 25+23 is 48. 48 is not divisible by 13. The whole number is not divisible by 13.
I am getting tired of all this calculation, and considering giving up and going to the Wolfram Engine. Calculators are much faster at this.
I am not a computer, so I will.
One last try first.
23 x 100=2300. One more 23 makes… 2323. 101 23s get us there. This is like starting at 23 on a number line, and jumping 101 times. Or, even faster, starting at 101, and jumping only 23 times.
23 x 101 is the prime factorization.
It was there all along. Assuming 101 is actually prime. We don’t have to check past 10 (101 is just above 10 x 10), so all I am worried about is 7. It is not divisible by 7. 98 and 105 are the nearest multiples of 7.
Think about that rectangle. It is about 5 times as long as it is wide, but it is not nearly squarish. Actually, it is quite like the (nearly Platonic) rectangle that kids come to picture through worksheets and instructional materials-you know the one. Can you see it?
2323 has two possible rectangles:
1 x 2323 (the long cigar)
23 x 101 (a fairly standard looking rectangle).
Here is a website you can ask if numbers are prime, if you resent doing all that work:
Is it odd that 23 is a factor of 2323? Or was it more likely than not? Number theorists, weigh in.
Also, what is an interesting factorization you know and like?
