This may not be everyone’s (anyone’s?) cup of tea, but recently Derpetologist mentioned in the comments that he had noticed a certain property of prime numbers. I looked at it for a while and came up with a proof. R.J. suggested that I write an article on it, and Tonio was agreeable, so here we go for those who might find it interesting. If not, feel free to skip to the comments.
Derpetologist’s conjecture: The difference of the squares of any two prime numbers 5 or above is divisible by 6. (Whiz’s corollary: said difference is also divisible by 24.)
I worked out a detailed proof of the conjecture (and my corollary) using the fact that they are divisible only by 1 and themselves. But then I wondered if there is any profound significance to this property of primes. Is it in some way more general than just for primes? It turns out the answer is yes and it relies on the fact that in order to be divisible by 24, a number must be divisible by both 8 and 3. This led to two, more general, theorems, which when combined indicated there was a larger class of numbers that guarantees divisibility by 24. Note that all variables used below are integers.
Theorem 1: The differences of the squares of any two odd numbers is divisible by 8.
Proof: Any odd number can be written as 2n + 1. Therefore the difference of squares of any two odd numbers (not necessarily prime) is (2m + 1)^2 – (2n + 1)^2 = 4 (m – n)(m + n + 1), which is clearly divisible by 4. Furthermore, if m and n are either both even or both odd, then m – n is even and the difference is divisible by at least one more factor of 2, or, in total, by 8. If one of m and n is even and the other odd, then m + n + 1 is even and again the difference is divisible by at least one more factor of 2, and the difference is divisible by 8. Q.E.D.
What about divisibility by 3? Is there any significance to the fact that the prime number 3 is excluded from the set of primes included in Derpetologist’s conjecture? It turns out that the answer to that is yes.
Theorem 2: The difference of squares of any two numbers that are not divisible by 3 is divisible by 3.
Proof: Any number not divisible by 3 (not necessarily prime or odd) can be written as 3n + 1 or 3n +2. Then for two different numbers of the first form, the difference of their squares is (3n + 1)^2 – (3m + 1)^2 = 3(n – m)(3n + 3m +2), which is clearly divisible by 3. Similar computations can be made for any other combination of two numbers not divisible by 3, either one of the form 3n +1 and one 3n + 2, or both of the form 3n + 2; fleshing out those cases will be left as an exercise for the reader (!). Q.E.D.
Proof of the conjecture: Since all primes 5 or above are both odd and not divisible by 3, by Theorems 1 and 2 the difference of their squares is divisible by 8 and 3, respectively, and therefore also divisible by 24. Q.E.D.
Thus the original conjecture and corollary are true because all primes 5 or above are both odd and not divisible by 3, but other pairs of numbers that are not prime, but are odd and not a multiple of 3, also have the same property; for example, the difference of the squares of 35 and 25 is 600, which is 24 times 25. So primes 5 and above are just a subset of odd numbers whose difference of squares is divisible by 24.
It seemed to me that this property of primes was simple enough that it must have been known for a quite a while. After completing my proof, I looked for others, and found a different one that used Fermat’s little theorem (which I won’t describe here — you can look it up if interested). The other proof, however, was good only for primes 7 and above.
We are so much more intellectual than other libertarian chatrooms.
I was un/reliably informed this weekend that libertarians are just Republicans who are ashamed to admit it.
It was but a decade ago when progressive was still considered a dirty word that Democrats and leftie cunts everywhere ran from. They took it back and embraced it. Same with socialism.
Not particularly relevant, but just an interesting and sad note from history. Obama unfortunately did fundamentally change this country.
Was it “Republicans who want to smoke pot” or just a new take on the old refrain?
I think it’s just a re-recording of an old hit. Maybe with a bit of new arrangement.
And that damn Auto-tune! Blech! 😝
Yes. How great is it to have a group that can discuss everything from fart jokes to advanced math proofs. Once again, Glibs wins.
Square roots, multiplication, division, fractions…all of this is just blasphemy against the First.
Those points of ellipses doomed you.
GAY!
Of course, the good book does say: Multiply and be fruitful.
So I shouldn’t be surprised.
* Seriously, nice article.
I thought it was be fruitful and multiply.
What’s the order of operations on these things?
*eyes glaze over*
How does this help me win the lottery?
I used to work with a guy who graduated with a math major. He was not only the first in his family to go to college, he was the first to leave the trailer park.
He would complain about family reunions because all of his kin were convinced that as a math major he knew some secret to winning the lottery. The would pester him constantly for secret tips that only a math major would know.
Finally he decided that he would come up with a tip to get them off his back. I can’t remember the exact formula, but it was basically if you had to choose numbers between 1 and 100, you wanted all the numbers you picked to average out to 50. He gave them some big song and dance about if you looked at the history of the lottery the winning numbers averaged out to 50.
His family was ecstatic.
However, the honeymoon didn’t last long. Not because they didn’t believe him and his theory. Nope, his new problem was them constantly calling him to help him calculate the average of the numbers that they had picked. And how to update their numbers to get to that magic target of 50.
I have a crush on higher math. I found Derpy’s conjecture to be very interesting and I thought about trying to prove it myself, but I can’t because I don’t understand higher math. I aspire to higher math.
As proofs about properties of prime numbers go, this one is fairly easy and doesn’t require various esoteric theorems.
If there are numerals involved, it’s probably not higher math.
I’m usually working during the day, but every so often I get a break and check out the midday post. Thanks to a trip to the ER due to a kidney stone, today is one of those days. Morphine and math don’t mix, but I think I’ll check out yesterday’s SF post, that should be a hoot. I’ll revisit this when my head clears, thank you very much.
Oh, dear! So sorry, ‘patzie! 😖 Feel better quicker!
Same kidney stone working its way down or a new one?
Glad you got the good stuff!
Thanks, y’all. New stone just beginning the journey. 4mm, should pass on its own.
ER doc is clueless. Tried to send me home with a prescription for a strong NSAID despite the fact that I am on a blood thinner for clotting issues. When I pointed this out, he changed it to hydrocodone which won’t cause bleeding issues. The medical profession is in sorry shape nowadays.
Yeah, but what was their pronouns?
Doesn’t matter, xir shoulda known better.
Based on the teacher story from this morning’s links, I’m guessing Dx.
That seems derivative.
Feel better!
For some strange reason, in recent years I’ve found myself compelled to put certain numerical settings like volume levels at prime numbers. Is there a scientific term for such a psychological condition?
OCD?
Don’t worry, I get twitchy when the volume setting doesn’t end in a 0 or a 5.
No love for eleven?
Not when I can go to 15.
Do you have to eat your skittles in order of color, ie. no two of the same colors in a row.
For any candy that comes in an array of colors, you form a series of sets, then eat the remainder before going set by set. The remainder may contain contiguous colors.
So we agree.
You sort by color, then eat from the largest group until its the same size as the next largest, then alternate until same size as 3rd largest, and so forth.
Incorrect.
Sort by color.
Then eat from least favorite to most favorite color.
Depends on if I’m eating them out of the package, or if I’ve dumped them onto a surface to sort them. Out of the package, I don’t care, if I’ve dumped them out, I will sort them.
Also this.
You people are sick.
*ADHD person logs out*
(Note to self: avoid listening to recorded music or watching TV/video with Mr. Ilium without the presence of an independent arbitrator.)
My amplifier doesn’t use a 1-10 scale; it goes from -80 dB to “Holy-Shit-That’s-Loud!” dB (plus “mute,” of course).
I find myself selecting volume levels in 2.5dB increments.
I won’t turn the volume to a prime number.
11!!!
42!!
NOT PRIME!!! ::hyperventilates::
Neither is 11!!!
Which is an insanely large number.
<a href="https://tfwiki.net/wiki/Trukk_not_munky"
Sigh… well, you get the joke hopefully. And that my tag skills are a further joke.
My girlfriend abhors odd numbers: her cocktail much have either 2 or 4 ice cubes in it; never 3. The TV volume has to be 18 or 20, never 19.
That’s a bit odd.
So no 69, huh?
Can we get the #1 ass slap gif?
But he can get the 68 and owe her one.
Five is right out.
So its either mico-penis, standard or porn star for her huh?
That flies in the face of cocktail traditions. For garnishing a martini, one should use 1 or 3 olives, an even number is supposed to give bad luck.
7 is the magic number it seems. Even if you are more likely to operate in even numbers. We humans (or maybe just Americans) love the #7.
Jack Daniels, Snow White, soda, sins, rainbow colors, wives’ tales, our most primal memory capacity (phone numbers), etc.
This amp goes to eleven.
I do that when warming stuff in the microwave.
Need to warm for abour 40 seconds? 41 or 43 work nicely.
44 is more efficient.
Subpriminal messaging
Wait, aren’t all numbers devisable? Do you mean devisable by whole numbers? Or have calculators been lying to me my whole life? Is the FED in charge of Big Calculator? Texas Instruments…TI…..Tax Involuntarily…
Yep the math checks out.
No numbers, so it’s higher math, also!
We’re talking integers here, of course. But you knew that.
Gee, whiz, that’s the reason I was a liberal arts major…
Since it appears no one else has
Great article (if a little inaccessible for hockey damaged brains!)
Exactly. Now where is my #2 pencil and graph paper
WaPo Fact Checker: Biden doesn’t lie. He just tells tales that aren’t credible. https://www.washingtonpost.com/politics/2023/08/31/biden-loves-retell-certain-stories-some-arent-credible/
Ah, so he’s a bullshit artist.
As opposed to the previous guy whose bullshitting is a threat to Our Democracy.
That is some wordsmithing. So now we have ‘credible’ versus ‘truth’. Fucking hell.
Fake but accurate
Truthy?
truth > facts
Fucking Kessler, of course.
proof: Any number not divisible by 3 (not necessarily prime or odd) can be written as 3n + 1 or 3n +2.
Am I interpreting this correctly?
How can you write the number 4 as 3n+2? 4 can be written as 3n+1 (3*1+1).
The difference of the squares of any two prime numbers 5 or above is divisible by 6. Does this statement also apply? If so then my bad.
Yes, if it’s divisible by 24, it’s also divisible by any number that 24 is divisible by, so 2, 3, 4, 6, 8, and 12.
I thought it was 3n±1
It’s or, not and, i.e., one or the other and not necessarily both, so 4 = 3(1)+1.
3n +2 is functionally equivalent to 3n-1 as far as integers not divisible by 3 is concerned. Or are you just being difficult? 🙂
This math doesn’t add up.
https://www.reuters.com/legal/former-proud-boys-leaders-could-face-longest-sentences-yet-us-capitol-attack-2023-08-31/
https://amgreatness.com/2023/08/30/denver-to-pay-5-million-to-blm-rioters/
((biased system) (unsympathetic defendants)/leftist judges)) = FYTW
Works for me, you must have forgotten to carry the 1.
UnCiv: Don’t get too close to Milwaukee
https://www.tmj4.com/news/local-news/oconomowoc-playground-closed-due-to-a-mouse-sighting-investigation-underway
I’ve thankfully been free of the beasties for some time. Ever since I found the point of entry.
Imagination Station Playground is Waukesha County’s first fully inclusive playground. It is located in Roosevelt Park adjacent to the Lake County Trail.
As the City works to kick the little guy out, officials urge the public to explore its many other playgrounds.
Yes, but are the other playgrounds fully inclusive? Do they allow dogs and the Irish?
Doesn’t fully inclusive mean that food and alcohol are included in the price?
No, that’s “All-Inclusive” Different, totally different.
Apparently not inclusive of mice. 🐀
Beat me to it.
They are closing an outdoor playground because of a mouse? I can understand avoiding Mickey Mouse these days, but this is ridiculous.
Didn’t anyone tell them there’s 22 more letters in the alphabet?
Sounds like a town in New Jersey
Who wants to go for a hike along the Musconetcong River in Hopatcong?
Cant, I have an appointment in Buckshutem tonight.
It is Iroquois for “Cat Dances on Keyboard”
Damn. This is terrible.
It’s a crock of shit that could be covered by a simple vandalism charge. The legal system is broken beyond repair.
Page is down.
summary please?
Joe Biggs, one of the J6, just got 17 fucking years.
fuck
And that’s a Trump appointee as well as a member of the Federalist Society.
Choke on that one.
Feds are always feds first, ahead of any other consideration. Behave accordingly.
The Federalist Society is rather jack assed. Their dedication to the letter of the law is embarrassing at this point. Law is nothing more than elite morality, which is why Eugene Debs gets thrown in jail for opposing WWI, while Abbie Hoffman gets off scotfree for the same invented crime just thirty years later (without overturning the Debs decision). Jack assed “constitutionalists” will pretend as if this was the court correcting an error, but they did no such thing. The only difference between the two is that Debs opposed a war beloved by the elites whereas Hoffman opposed a war that the elites hated (at least once they got rid of the college exemption for the draft).
Federalist Society judges are ideologically possessed with the asinine concept that the law is something sacred and eternal, which it clearly is not.
“Reward friends, punish enemy” is the only eternal law in civil society (it sure as hell isn’t ‘free speech’ or ‘free exercise’ as both have been violated repeatedly by the courts in the recent past).
Just like Alito and his jackass invisible ink is ok as long as it’s from the time the constitution or amendment is ratified, not there’s a plain meaning to the text and if you don’t like it, use the amendment process to change, clarify, or limit it.
“Government property”
GFY
Not you Tundra. Or even the article writer. Sadly that is accurate.
summary please?
Trespassing and vandalism are now High Treason.
I assume that quoting the Declaration is now Sedition and Treason.
Also, see: Gadsden Flag or other patriotic symbols that DHS deems to be those of domestic terrorists.
And Latin Mass… and being pro-life… and anything Not Of The Body at this point.
Looking Trumpish…
Francis the Coward has nothing to say but CIA talking points
To be fair, the billionaires at Davos needs a pope too, just not a Christian one. Good thing Satan made Jesuits
I assume that quoting the Declaration is now Sedition and Treason.
That depends. Whose side are you on?
Does it matter?
<>
*sigh*
too lazy to quote old Russian joke about three guys in prison
This one?
One of them asks the two others: “So what did you do?”
The first one answers: “Well, I arrived late at the factory, and so they accused me of slowing down the Revolution and the victory of the Proletariat.”
The second one answers: “Well, I arrived early at the factory, and so they accused me of wanting to be favored and promoted over my fellow workers.”
Then they turn to the one who asked the question: “How about you, then?”
“Well, I arrived at the factory right on time, and so they accused me of having a watch from the West.
One of many I suppose.
Per google:
3 men talk in prison:
– I was jailed for supporting Radek.
– Strange, I was jailed for not supporting Radek.
And the 3rd one says:
– I am Radek.
They are closing an outdoor playground because of a mouse?
Dude, those things are enormous, with antlers like snowplow blades. And if that’s not bad enough, they can be crabby as fuck. You never know what they are gonna do.
“Dat’s a tousand pound moose!”
As long as it’s not a kangaroo mouse.
https://m.youtube.com/watch?v=VV0eGJT4_Mg&pp=ygUbc3lsdmVzdGVyIGNhdCBhbmQga2FuZ2Fyb28g
Oh, Father.
Don’t forget the ticks.
Re Skittles. You collate them into sets of the flavors. You eat one set at a time until all your sets are gone, and then you collate the remainder until you have no sets left and all there is is one lonely grape Skittle.
How do you people eat boxes of assorted chocolates? What’s the strategy there?
If it comes with a key on the underside of the lid, I identify the junk and make sure I don’t get stuck with those.
Meh… I eat two at a time, matching color (because matching flavor) obviously… whichever color I feel like next.
Anarchy! Anarchy!
grape Skittle.
Ewwwww.
I am disappoint. The eating of skittles has exposed the grim OCD underbelly of Glibertopia.
You just pour a handful and put them all in you’re mouth at once.
Incorrect.
You throw those evil carbs in the trash and get some jerky.
Sadly, this is where I am in life. I am in mourning, I tell you. It’s as bad as it was when the lime turned into green apple.
*me, eating Skittles PROPERLY*
*post-Skittles clarity: pounding headache for days on end*