Six months running I worked alone in my garage on primality. Used the topic to divert my thoughts,, while trying to fall asleep from some seriously depressing side-concerns, several of which I’m actually forbidden to discuss here.
Factored every number in turn up to 2000 and beyond in my head, developed tricks for dividing by every number under the Sun.
The point is that I knew that I had a Goal, I just felt it in my bones. Resisted boning up on Fermat, Euler, Poisson, all the gang who’ve been obsessed on the question for millennia. Somehow I knew that my approach was.. emotional?
Early on I ‘got it’ that if 41 is prime here on Earth, Rock 3 from the Sun, then it was also prime in the Pleiades. This is oddly and truly comforting to realize. You got 41 pebbles on the beach in an alien star system? Well god help ’em, even the smartest ETs can’t line ’em up in tidy rows and columns. We are not alone? At least in our misery.
And with all respect to my older son’s claim, Primality has ‘nuttin’ to do with choice of number-base system. I finally agreed to disagree, told him to ‘Enjoy yer world where stuff is, like ‘different’, and stopped arguing.
And now to Emotions.. Feelings:
Um I feel a deep dislike, hatred even, of, for example, ’61’, that sad-sack number. (God should wipe out his Name and Memory). To my mind and heart there are four Prime Numbers, as per the title: 2,3,5, and 7.
All the rest are losers, ne’er-do-wells, and misfits.
Now it’s not rocket science to discover that lots of primes occur as pairs ‘Twin Primes’ as they are called. Almost always on either side of a nice number divisible by 6. Like, say 42,(six times seven), surrounded by 41 and 43, evil little failed numerals.
Let’s look at 61 for a moment. He lives right beside 60, who has almost nightly parties on the lawn with his many loving factors: 15, 12, 10, 6, 5, 4, 3, and 2. Hot dogs and buns, who cares that they’re sold in 6-pax and 8-pax, he’ll get by. He even made friends with Heinz ketchup (57 kinds) after he realized that 57 is really just 3 sub-flavour-groups with 19 varieties in each group. Yeah, even in the Pleiades.
So 61 is jealous at root, and spiteful to boot. The side-porch he built toward the other neighbor, ’62’, gave him less ‘misery loves company’ consolation than he’d hoped. ’62’ entertains 31 and the busy-but-party-animal 2 almost every week-end… while 61 hides in his stupid basement , alone with a cheap beer and some pirated math-porn.
I have zero sympathy, it turns out. They made a choice, now need to live wid it.
Eleven, you might say, why not include eleven?
Good question, and here’s a good answer: Of all the numbers up to 2000, eleven ‘owns’ a pathetic three integers: his own sorry ass, plus 121 and 1331. All the rest belong to other ‘real-men’ primes. I’ll explain:
First of all, ‘2’ grabs fully half of the natural numbers, the even ones, duh.
Then 3 goes out to claim a third of them, but comes back, tail between his legs, with one-sixth. Half of his ‘clients’ were already claimed by 2, like 6, 12, 18, 24, etc.
Five’s turn happens, and his dream of twenty percent kinda melts like an ice-cube here in the Israeli summer: Numbers ending in ‘0’ belong to 2 already, and of what’s left, say 15,25,35,45,65, etc, well, ‘3’ already grabbed a third of them.
Seven has the shittiest deal, one could say. Every one of his homies has been tagged by some other gang, save 49 and 91 (of numbers up to one- hundred.) Either they’re even, or divisible by 3 or 5. Ninety-one is where he hides out and built his embassy. But only because Johnny ‘Pluto-ized’ 13. (7X13=91)
So, enough already. I don’t give a rat’s ass about ‘percentage of primes approaches asymptotically n/log-n.’
Like in any social setting, even Word-Press, once you decide who your real Friends are, the rest is stamp-collecting. I’m ‘friend-ing 2357 and disabling ‘Requests’.
I do hope anyone reading enjoyed this romp. I just tell the truth, like Truman; other folks call it number-hell/ JS