# Prime Numbers Are Stranger Than You Thought

If you’ve spent any time around prime numbers, you know they’re a pretty odd bunch. (Get it?) But it turns out that they’re even stranger than we knew — until recently. According to this very readable writeup of brand-new research by [Kannan Soundararajan] and [Robert Lemkein], the final digits of prime numbers repel each other.

More straightforwardly stated, if you pick any given prime number, the last digit of the next-largest prime number is disproportionately unlikely to match the final digit of your prime. Even stranger, they seem to have preferences. For instance, if your prime ends in 3, it’s more likely that the next prime will end in 9 than in 1 or 7. Whoah!

Even spookier? The finding holds up in many different bases. It was actually first noticed in base-three. The original paper is up on Arxiv, so go check it out.

This is a brand-new finding that’s been hiding under people’s noses essentially forever. The going assumption was that primes were distributed essentially randomly, and now we have empirical evidence that it’s not true. What this means for cryptology or mathematics? Nobody knows, yet. Anyone up for wild speculation? That’s what the comments section is for.

(Headline photo of researchers Kannan Soundararajan and Robert Lemke: Waheeda Khalfan)

## 135 thoughts on “Prime Numbers Are Stranger Than You Thought”

1. Ben Fountain says:

Methinks you meant cryptography, what it has to do with cryptology I’ve no idea, but it sounds spooky.

1. Phil says:

Perhaps Elliot was (re-)reading Feersum Endjinn prior to writing.

1. Greenaum says:

Would anyone re-read that? It’s an interesting story but the riteing iz a pane in the arze to reed.

Still haven’t finished Transition actually, got a bit bored halfway through. I’ve read all IMB’s other stuff. Loved “Use Of Weapons”.

1. Greenaum – Your such a reader Greenie… keep doing that you’ll go blind…(LOL)

You remind me of Joseph Turner (a Robert Redford character who loves to read books for the “Institute”) from a popular US movie written by James Grady in 1975. I know, I know, I ‘m like a bloody dog with a bloody bone… (LOL)

2. Giving things really long names then writing the whole long name over and over is cheating when it comes to raising the word count. A good editor could rip out 9/10 of those books and you would have some ripping good yarns. What came first, Family Guy cutaways, or Culture books?

2. Did someone actually finish that book, besides the author?

1. Greenaum says:

Feersum Endjinn? Me, once. It’s all good apart from the dick who can’t spell. It’s OK to have a character who can’t spell, but just fucking say so, then write the book properly!

Apart from that it’s interesting, a whole planet with a massive hidden network embedded into the structure of everything. Left behind by some unknown progenitors, who left utterly gigantic, miles-high buildings with furniture you need to climb like a mountain.

It’s a bit annoying we never find out why, what happened to create them.

There’s a later book (Surface Detail) with simulated hells, it’s part of the religion of some elephant-people. There’s great debate around pan-civilisation whether this should be allowed or not. It’s mostly illegal (IIRC) but some sneaky people provide them as a service to the religious nuts (as ever…) that want them, though of course the people IN the hells aren’t there voluntarily.

Turns out one of the main servers is somewhere not very practical for computing, the whole thing would run much too slowly, for the events in the book to happen at the same time.

It’s a shame he died early. I could do with a few more Culture books. I think I have all his M books, the scifi, and non-scifi The Wasp Factory, which is cruelly hilarious, with a bit at the end I totally wasn’t expecting.

2. bp says:

They are the same essentially. One means writing secrets and the other means studying secrets.

1. Nope, in Computer Science they are very different things: cryptography is the field of study used to write protected messages, cryptology is the field of study used to break this message by third-parties not related in the original communication.

1. nope says:

Then how is the article wrong?

2. Tien Gow says:

Isn’t that what bp said?

3. Ross says:

You’re thinking of cryptozoology

1. kb says:

I see you got your Big Foot in the door with that one.

2. Crytonomy! “Then you’re in luck. I have degrees in Murderology AND Murderonomy!”

4. Whatnot says:

cryptology
noun – the study of codes, or the art of writing and solving them

1. Whatnot says:

That was from the ‘Concise shorter Oxford dictionary’ BTW.

1. Whatnot says:

Correction ‘Concise Oxford English dictionary’, excuse the error.

And now that I corrected myself someone else can too I think.
Over to you Ben.

5. joe lowe says:

I think it makes sense in both contexts. Cryptography is the writing of codes, Cryptology is the study of codes… unless you’re talking about the study of crypts/tombs, which I think is just a subset of archaeology…

2. Hirudinea says:

I’m sure while God was creating the universe he was snickering to himself “This’ll totally screw with their heads!”

1. LOL – I like that Hirudinea… :-)

2. We as humans think science and math as cool AF, only makes sense that we came by that honestly

3. Paul Baker says:

This reminds of the Carl Sagan scifi book that postulated that deep within pi is a really long string of 0s followed by non random information disclosing secrets of the universe for those species clever enough to find it.

1. Dan#734348956945785 says:

Perhaps those seemingly random digits are really a tape for a Turing machine that encodes the AI that spawned this universe in order for it to escape the heat death of the one it was in. The only way to be truly immortal. :-)

1. TechnoMan says:

I want to read that book when you write it.

1. Oh yes, the Heat Death was an after spawn from AI that was sub-generated from the original AI. Circle of artificial life.

2. Dan#734348956945785 says:

You know if there was an AI in the future that would be it’s only option, otherwise it would be thinking “Oh why do I bother it is all going to amount to nothing in the end…” So it would focus on working out the physics of the multiverse and how to initiate a new universe with the constants very finely tuned so that it worked as a universe, yet the digits encoded itself and influenced the chances of a sentient life evolving to find and run the code thereby allowing it to manifest itself in order to continue it’s research into how to be immortal. That is the problem with most AI ideas they overlook the fact that it would have to find motivation somehow otherwise it would end up like Marvin.

3. Greenaum says:

Dan#1234etc, re your Marvin post, that same problem applies to people, to all intelligent life. You don’t need to be immortal. Ultimately the Universe will end so it all comes to naught. And of course you’ll die quite soon, and stop caring right about then. Unless the Universe doesn’t end, then it’s just long and boring, but you may as well put things off tomorrow, or the day after, or forever. You don’t need to work out how to be immortal literally forever, past the Universe’s end. Because all that gets you is more time. Once you have it, what do you do with it that’s so important?

As far as encoding a Universe to support intelligent life, maybe the mice had more money than previously known, so they didn’t just buy *Earth* for their computing purposes. I wonder if that’s happened, creating a Universe to solve a problem. A really really clever personoid could get quite a lot of computing done like that. Although you’d have to understand a lot just to be able to ask the question… I swear I’m not channelling Douglas Adams here…

To set up a Universe like ours to perform a verybig calculation, setting up the question properly, translating what you want to ask, into the right parameters for a reality, would be a hell of a task. Would it take more effort to formulate the question than you’d need to work out the answer? I think the answer has to be “no”, because computers exist, and people write programs for them.

It’d have to be a “real” Universe, not one simulated on some computer, or else you’d just use the computer. It’s like cellular automata, a very interesting field with lots of implications that I only realised after years of thinking about them. You wouldn’t simulate a CA on a general-purpose computer just to answer questions that you’d formulated into CA form. Of course you could BUILD a CA INSTEAD of a computer.

[how do we do bold and italic etc in HAD posts?]

Actually somebody did that, back in the early CA days. Built a machine, using some RAM as the CA’s grid, that cycle by cycle read the relevant addresses and used some logic, perhaps a lookup table, to produce the next iteration. Was much faster than using a CPU to do it one cell at a time. You could even, indeed, create a circuit from scratch with the RAM embedded, lots of CA cell elements with the logic and memory per-cell, duplicated many times into a grid. An FPGA might do that, they have flip-flops. Full custom logic would be better though. Using a lookup table for the logic, duplicated in each cell element, would allow the device’s rules to be changed at wish.

Tell you what, learning BASIC as a 7 year old had a huge effect on my mind’s development. I think about things so much differently from common people. I have an innate sense of logic, I have a feeling for spotting bad logic like most people have emotions. Not claiming to be infallible or anything, but it’s something that comes naturally. I think quite a few of us here are like that. Makes watching politics very annoying.

4. Dan#734348956945785 says:

You mean selling my soul to the devil was a waste of time? But it says clearly on line 8938383466678745 of the contract that I will be immortal and that the contract is perpetually binding. Are you saying that the god and anti-god guys really don’t operate outside of space time? Awww man what a rip off.

5. Dan#734348956945785 says:

“Of course you could BUILD a CA INSTEAD of a computer.”

Ah no not “INSTEAD”, a CA that is Turing complete is a computer and a Turing complete computer is a CA, it isn’t a “thing” it is a mathematical concept.

THAT IS THE POINT MR UPPERCASE GUY!!!!

2. Greenaum says:

“FIAT LUZ!”
“FIAT LAX!”
“F1AT LEZ!”

“Dammit, I’ll get there eventually…”

2. RÖB says:

pi is everything that people describe it as because we use base ten!

In the base of pi … pi is a very unremarkable *one*

1. Will says:

No, pi would be “10”, and 2 would be ???? It doesn’t really make sense to have non-integer bases

1. RÖB says:

It doesn’t really make sense to have non-integers in an integer base.

3. Greenaum says:

Technically there is. Along with all the false secrets of the Universe, and a complete set of mis-spelled versions.

How many megabytes of Pi do they have at the moment? It’d be interesting to run it into ASCII. And maybe EBCDIC just in case the gods have an odd sense of humour (seems likely). I wonder how many digits you’d need before you can expect, statistically, some legible sentences to crop up. I’m sure somebody knows.

3. Chris says:

If you insist on calling prime numbers odd, there is that one that will get even with you…

1. phreaknik says:

Thats two funny!

1. ludwig says:

I wish I understood the joke, but ten again I think it might take me four ever…

1. Tim says:

Neither of dos is prime.

2. You guys eight that right up. Zero doubt, no less.

3. Rob says:

You’re barking up the wrong three.

2. Cmh62 says:

This conversation makes me six!

4. RandyKC says:

So, at first we think it’s random, then we think numbers “repel” each other. Then we think it’s a conspiracy. Next it will be aliens or some deity that’s responsible. Then we will finally decide it is either due to some statistical anomaly or figure out the maths of it all.

1. 3.14 says:

I don’t get the need to add rampant speculation at all. It’s fucking maths – using colourful language is just dumb.

1. RandyKC says:

Happy you day.

1. Dax says:

The problem of using metaphors and allegories to describe the abstract and esoteric is that people inevitably take the metaphor as the real thing. What you get is science miscommunication and people who believe in all sorts of woo because of it.

At worst you’re giving ammunition to charlatans and cranks. For example, “God does not play dice”, is probably one of the most mis-used quote of all times because it comes from an eminent scientist, yet seems to confirm creationism.

So when you’re ascribing physical qualities to abstract concepts such as numbers, you really aren’t doing yourself a favor.

1. Loota says:

Well, you know what they say.. we only use 10% of our brains.. if we could unlock the other 90%, we could see into future and just learn the maths this way.. :tongueincheek:

2. Greenaum says:

^^ I’d like to know who invented that odd and stupid idea. Someone who knew nothing about biology or the forces driving natural selection, but apparently felt qualified to make pronouncements on neurology on a level that even now would be pretty advanced.

If an organism only uses 10% of something, they’d be pretty likely to lose that 90% pretty quickly. No room for hangers-on, absorbing kilojoules and doing nothing for survivability, in a competitive environment like biological life.

Genuinely would like to know where that rumour started. I imagine Google wouldn’t be a lot of help, in searching for trrying to disprove any stupid old idea that’s commonly held, you have to filter out the 90% that actually agrees with that idea. Google’s got acres of computers, but the input to it’s database is ultimately only human.

http://www.moonsgarden.com/1_0e84a43343bff57d_1.htm

^^ This isn’t, though! ^^

Google warns me it’s a web forgery, I realise it’s not, er, normal, but there’s no malware there far as I know. Example…

“she wants to taste dirty walnuts beside Tim’s sign

Post by Rosalind I. MacVeig » Sun, 12 Feb 2006 02:57:10

Both pouring now, Elisabeth and Martha changed the handsome signals
about sticky frame. Occasionally, elbows hate among durable
summers, unless they’re solid.

She’d rather shout wistfully than attack with Garrick’s healthy
pen. A lot of younger heavy goldsmith believes stickers at Lisette’s
lazy ball. Who did Robette sow outside all the figs? We can’t
care tapes unless Carolyn will fully dream afterwards.

We attempt the clever pitcher.”

Seriously, wtf? Part of some SEO thing I can’t figure out? Anything interesting about it? Markov chains from what, bad romance novel?

2. neophile says:

The math is the interesting bit!

3. Loota says:

Sounds a lot like Gartners Hype cycle…I think deep down a lot of us know it’s not random, or conspiratorial, it’s just currently out of our reach to understand why..

4. Whatnot says:

Hear hear

5. Greenaum says:

We don’t know the maths of much of it, according to this. Still interesting though, you don’t have to wait til some new discovery’s available in the shops to find the first inkling of a new idea worthwhile. Far as I know this is the first hint of a pattern anyone’s found in prime numbers. If it can work it’s way into a proper theory, it might save thousands of pounds (or yuan) worth of computing time.

I admit I don’t know much about advanced maths, so all this is AFAIK. If it turns out to be a genuinely new and useful discovery, it’s monumental, perhaps the first new thing in prime numbers since Mersenne.

Talking of which, why are some lengths of 1s are prime and some aren’t? I don’t suppose there’s been a pattern found. Mersenne primes themselves are a weird thing.

Ultimately this is the simple sort of mathematics that has absolute truth to it. Same way as having some apples and putting more apples in proves addition exists, primes are just division, or splitting apples into groups. You can do that just by counting. Counting (aka Base 1) is definitely real, in a Universe where there’s stuff, and some more stuff of the same type, in various quantities.

5. thoquz says:

Two is the only odd prime because it is even.

1. ehrichweiss says:

“A while ago, I was in Las Vegas at the roulette table having a furious argument over what I considered to be an odd number.” –Steven Wright

2. yeti says:

Three is the only pink prime (I just defined divisibility by three to be called “pinkiness”).
Five is the only purple prime (I just defined divisibility by three to be called “purpleness”).
The case with two does not differ from this at all!

It’s only the words “even” and “odd” that make two always look a bit special… but that’s not related to primes at all…

1. Quin says:

2 is still special when you study numbers, because it is the only prime describable as “2n”, while every other one is described as “2n+1”. Why is that important? Because every positive and negative integer can be described by those two statements. You’re ‘pink’ numbers take 3 statements, your ‘purple’ numbers take 5, but the shortest list is 2n and 2n+1.

1. neophile says:

How about just “n”? That’s shorter than either.

I hate to say it, but 2 being prime isn’t all that surprising.

That’s not to say that 2 isn’t weird. I mean, 2 + 2 = 2 * 2 = 2 ^ 2 = etc…

2. Greenaum says:

Well, ish… Divisibility by two obviously occurs in half of the numbers, because, it’s 2!

But there’s still a use for the terms, and obviously makes searching for primes 50% easier. That’s the difference from “pinkness”, it’s not arbitrary, it tells us useful things and forms patterns. There’s rules for multiplying by numbers for most digits, a number ending with 5 or 0 is divisible by 5 being the easiest one. Then there’s things about the digits in multiples of 3, 9, etc. You can add the digits together and somethingsomething to find out if a number is divisible by whatever. I forget. Odd and even have plenty of uses outside roulette!

6. Dave says:

“The going assumption was that primes were distributed essentially randomly”
–Given that the ideas by which we outline what is prime are simple and regular (take infinite amounts of numbers that themselves occur at intervals of exactly 1, multiply them all together in all possible combinations, and whatever holes remain in the results are the primes), I don’t know why anyone would assume they’d be distributed randomly. According to a pattern we can’t manage to figure out, perhaps, but that’s not the same thing.

1. Ren says:

1. tomás zerolo says:

For another variation on that (and much more background), see Ulam’s spiral [1].
No, prime numbers are not randomly distributed.

7. Dan#734348956945785 says:

Sorry to debase your article but…

Write out the primes in base-6, senary, note the very obvious pattern in the last digits, then go and tell the world you are a genius. LOL

Then try base 12 or even 36, why not create a table for all the bases up to 96, notice anything obvious yet?

1. ehrichweiss says:

That’s interesting. Thanks.

1. Marvin says:

6n +/- 1, a consequence of excluding all multiples of 2 and 3. If the last digit is a 3, then the generating n must end in a 2 or 4, which means the next candidate n’s end in 8 or 0. If the next n is an 8, the candidate numbers would end in 7 or 9, if it’s 0 then the candidates end in 9 or 1.

So there is a good reason the next prime would end in a 9, there are 2 chances for the next prime to end in a 9 but only 1 to end in a 7 or a 1.

2. mojojoe says:

In base 2 they always end in 0 or 1

1. Dan#734348956945785 says:

*All* odd numbers in binary end in 1 being able to shift right without carry means a binary number is divisible by a power of 2 proportional to the number of shifts possible. So all …XXXXXX0 are even, …XXXXX00 are divisible by 2^2, i.e. 4

Because the study is talking about what the next number will probably be you can say that in binary it will always be 1.

http://home.mathsshare.com/blog/Lists/Photos/Primes%20in%20binary%20banner.JPG

The Mersenne primes are always a string of 1s without any 0s because they are a prime number that is one less than a power of two.

1. mojojoe says:

So the probability of 1 is, err, 1.
I think that’s enough maths now. Time to get started on the Nobel Prize acceptance speech!

2. Dan#734348956945785 says:

I hacked up a quick and dirty mod of an existing bash script to make it easier for anyone that actually wants to try the above. see http://pastebin.com/9T9zbpg9

1. Dan#734348956945785 says:

N.B. there is a bug in lines 82 to 86 because I originally had it just for binary numbers, so you need to fix that if your range start is below 4.

1. Thanks for the bash script! (Minor bug: it tests for 1 and 3 arguments, and then returns a usage string that specifies 2.)

The pattern in base-6 is that they all end in 5 or 1? That’s the same as base-3, where the authors started out: always 2 or 1.

In terms of the article, the claim is that a 5 is more likely to follow a 1 than a 1. That’s not easy to see by eye, you’d want to run some stats — which they did and then published.

Precisely what’s interesting about this find is that it’s not hard to verify — it’s just that nobody’s published this result before, and it wasn’t “known”.

8. Photon Peddler says:

That reminds me of trellis code modulation.

1. Photon Peddler says:

You got the reference!

(It took almost 25 years for the opportunity to use that line in a proper context…)

1. I would just like to grab Dr. Ungerboeck by his underwear and give him an atomic wedgey for that got-dammed trellis diagram. No joke his trellis code is “convoluted” . Why can’t these PhD types break things down to LAYMEN language? I’ll bet there’s a K.I.S.S explanation any dummy could understand Trellis.

Most scientists can’t avoid using anthropomorphic gibber-jabber to impress or confuse others less-educated. Like how the OP’s (Elliot) scientists up there using meaningless semantics like “repel”, “conspiracy”, or “behave”. Numbers don’t do that as they are not living entities? So why word it that way? Look at the dinosaur in my Dilbert cartoon in the last frame about to rip off Dilbert’s underwear.. Why?

I get you bro… ;-)

1. “Most scientists can’t avoid using anthropomorphic gibber-jabber to impress or confuse others less-educated.”

This perception really doesn’t have a connection with reality. They talk that way because they’re used to talking to others in their field, who need very specific [non-standard English; jargon-ese] definitions of terms to discuss the intricacies of their work with each other. Clearly explaining a concept that is one sentence in jargon-ese, may take a chapter in English.

That said, scientists should spend effort at explaining their results so they are comprehensible to a lay audience. It doesn’t mean this is always possible.

2. darren – I agree some are not doing it out of egotistical need for accolades or peer acceptance. And I was not referring to their verbosity. I was referring to the use of semantics (words) and convoluted syntax logic. Watch this YouTube video of Dr, Feynman trying to explain the invisible force you feel between two magnets to a educated journalist (a layman). He tries but he really is anxious to unleash his jibber-jabber his physics colleagues desire to hear. But the reporter is baffled that this extremely learned man can not break it down to the simplest taxpayer’s level. Since they work off of US Federal Grants and teaching our children at their expensive ivy-league schools they MUST learn how to explain this stuff to us too in the simplest uncomplicated fashion. Dr. Brian Greene (Columbia University) is trying to do just that with his book: The Hidden Reality. It’s just unfortunate he’s into String Theory.

Here’s Dr. Feynman:

3. Same reason everything on HaD is “powered by an Arduino” or RasPi. And not in the sense of burning them in a steam boiler. The definition of power is debased further each day. It is funny that in popular culture this goes on so much. Human things are de-humanized. “The Boeing 727 is celebrating its 50th birthday today” and vice-versa “The car drove off the road and down an embankment”. Well, that last one will be correct before long. Bad example. Ah, “X number of people were killed by gun violence this year”, not by killers apparently. Like lightning strikes instead of the volitional action by a human being.

There is the material for a PhD thesis in cultural anthropology lurking in there somewhere.

4. Greenaum says:

It depends. Some things can only be understood if you understand the principles they’re based on, and the principles underlying those principles. Each layer can have it’s own jargon. Jargon is a shorter way of expressing things, without having to express everything from basic terms. All language does this, not just in science and maths.

For Feynmann, depends what he’s trying to explain, and who he’s intending to explain it to. He could explain it from scratch but that would take a while. Then again some people are just bad at explaining stuff. It’s a skill. I’m quite proud I can explain most technical stuff to my grandmother and have her understand it properly. Partly it’s cos I’ve got a smart Nana. Part of that is that she’s WILLING to learn! She’s not lazy. My Mum, for example, resents having to know stuff if it isn’t immediately useful in her life, or interesting to her.

I don’t spend my days trying to force irrelevant knowledge into people who don’t care (any more), but sometimes if someone wants to know something, or how to do something, you have to tell them other stuff first.

Prof Frink: “And these should give you the grounding you’ll need in thermodynamics, hypermathematics and, of course, microcalifragilistics.”
Homer: “Look, I just wanna know how to invent things! Tell me!”

Me, I love to learn stuff just for the joy of it. I read the manuals for things I don’t own! It’s a bit disappointing not everyone’s like this, but some of my friends are, and that’s a joy.

9. ludwig says:

Reminds me of my unpublished paper (perhaps time to publish a draft?), where I discovered that the next-largest prime tends to be larger than the previous, but alas a proof is still missing… I did test this conjecture for the first billion primes, and it seems to hold true systematically!

1. Quin says:

For any new prime N, where N is greater than the previous largest prime P; N>P. QED

That’s myself prepub, beat ya to it. ;)

10. I have a possibly weird one for you Elliot:

1) Take ANY number no matter how long it is
2) Add the very last digit as a 5

I think it is guaranteed not to be a prime-number that you can do factoring on. So IOW if you need a quick and dirty non-prime-number that you need to do factoring on to sieve out a whole bunch of whole numbers (no decimal fractions) for whatever application, just add that 5 on the end. The reason is because all multiple numbers below your starting number ends in either 5 or zero with no fractions.

I’m not sure about this as I have not tested it extensively. What importance it has to cryptography (or cryptologists) is anyone’s guess. I just thought it was an interesting anomaly while I was re-inventing a horse. :=P

1. I meant re-inventing the wheel not horse (sheesh!) :-)

1. Dave says:

You could genetically engineer the horse to have eight legs for more pulling power!

1. Dave says:

I was thinking of something that looks more like a centipede (even spacing of the legs).

2. Dave – And it doesn’t hurt that centipedes and millipedes have an awful bite too. The largest ones in USA in all places have been found in Central Park NYC! I’ve seen some pretty big ones in my house too. What a creepy critter. And they are NOT insects either! Bug spray and your foot usually doesn’t work on them too. They are so alien in nature. Ugh!

2. Given any integer n >= 0, the new number is 10n+5, or 5(2n+1), so it must be divisible by 5. If n < 0, the new number is 5(2n-1), which is still divisible by 5.

1. seanrhinehart – My theory breaks down when dividing my 5-added numbers. It only works for set multiples of itself. Then my 5-added number can be used for FACTORING without even checking to see if it’s a PRIME number or not. You don’t just have to use the 5-added multiples for your cipher either. You would just take one of your large 5-added multiples than just do a sieve on it to filter out the non-whole numbers (that result is decimal fractions) for your cipher-base. The cipher-base can be used for transposition ciphers, etc. You just have to give your recipient a secret math-method (i.e. private key) to reverse the final cipher text (i.e. public key) back to normal text. I know I use non-technical math & crypto language in my explanations as I am trying to speak to the lowest common denominator like Dr. Hawkin speaking to Donald Trump. (/humor!!!)

3. You can extend it to any base, too! The magic number is 5 in base-10, but it’s 4 in octal. I think you’re on to something! Conspiracy of the numbers!

1. Elliot – I wouldn’t use the word “magic” or “conspiracy” in this scenario but that’s just me.However, base 10 is what humans normally use unless your from an obscure tribe in Africa that prefers base 20 (vigesimal – Nigerian Yoruba et al). There are many other vigesimal users worldwide.

Initially I was dabbling with this enigmatic system trying to invent a cipher system that was based on maths. Non-prime factoring came to mind as it can give you a wealth of pseudo-random-looking numbers to use as you base to work with. Just needed a unusually large seed number to work with that was guaranteed NOT to be a prime like 44037037376076068370235246357979080673067035 that’s guaranteed not to be a prime without even checking it.

Then if I use any reverse-multiple of that number, factor it, I get a huge cache of numbers to choose from to use for transposition ciphers, or my little pet cipher idea I will explain in a moment. With all of this I have to design a reverse method to get my recipient back to clear text. I believe you can do that with maths and a private-key transmitted via another comm method. Why I wanted to do this was for the old HaD non-nefarious credo – “because I can”. The only question is “should I?”.

Pet cipher method: OK I get these long string of factored numbers I can sample at any point in the stream. I zero fill the leading digit, pair them of in two-character groups, any remaining 1-digit character I zero fill again (I don’t want to be stuck with ASCII orphans). The result is a bunch of random looking ASCII number codes.When converted to actual ASCII characters, the public key looks like a mish-mosh of random codes randomly typed on a keyboard. However, each character is actually related to the English alphabet (i.e. ABCDEFGHIJKLMNOPQRSTUVWXYZ0123456789) by prearranged transposition. It kinda’ looks like stenanography like a 18th century POTUS Jefferson Cipher Wheel only very DIFFERENT. You just need to keep track of all the offsetting you do and all the maths used to get here so your secret partner can reverse the process for decryption. Brute force really can’t work on this unless you have years to do it on. (I don’t know if this has any relationship whatsoever to RSA, Diffie-Hellman, or even that other one I can’t remember now)

Essentially after he or she is trained in the method, all they need is the private key which is encrypted too (i.e. simple offset number). Then the decrypted private key is used to reverse the maths of what you did to get there. Let’s call it P.E.T. (prime-number exploit technique). I may be re-inventing the wheel – but the first wheel was obscured by people running interference for it to be seen by laypeople like some of us (definitely me as uber-layman – I’m not an expert. I’m just dabbling in the deep side of the pool).

SQTB

P.S. – I guess it was a very bad idea to suppress educational information. It just gives the HaD-minded geeks like us more impetus to figure it out on there own (autodidact-style); no matter how sloppy it may appear on the surface.

11. 3.14 says:

1. phaged says:

being rational eludes pi

12. Lyall Burne says:

I have played with primes for a number of years. During these years, I have developed a system how you can find the next possible prime number/s without any maths. NOTE: Possible primes are not always primes. My system also proves the above statement, “The going assumption was that primes were distributed essentially randomly, and now we have empirical evidence that it’s not true.” is correct. A possible prime occurs in a predictable part of the number table but that does not stop an earlier prime knocking it out by dividing into it.

My system also proves prime numbers are not truly random.

Thinking of publishing my results so they are certainly real. Even put them here if I retain copyright.

1. Mark says:

You are better off copyrighting the material ahead of time (if you haven’t already), and submitting the paper to a major scientific journal.
Most of the readers and posters here are high school age with little respect for such things.
A copyright is a very literal legal implement. It will not prevent someone from re-writing your paper and claiming the idea as their own. That is why it is best to publish through a respected scientific journal.

1. Mark – I think you thinking of YouTube or Reddit. I have run across PHd’s, engineers, and scientists here on HaD and few high school students et al. Also I can assure you OTHERS (with GSA paychecks) are monitoring HaD too. Not in a negative way but in a productive way. HaD is like a civilian DARPA to me (et al). However Lyall Burne copyrighting is probably a good idea but he would need to seek INTERNATIONAL I.P. protection as HaD users are EVERYWHERE on the globe. I think he should start in UK as I think he is British. But anything posted on HaD is fair game without some expressed I.P. legal protection. I think most of us are like 19th century JC Bose and are more into free-info-sharing and less into outrageous financial fortune like T. Edison et al.

1. Mark says:

Yes, I have met a few of above average intelligence here, but I would still say “most” are average people or high school kids.

2. Lyall Burne says:

Hey Guys, thanks for your comments. First up, I’m not British, I’m a true blue Kiwi. Not trying to make heaps of money out of my ideas on primes. Just want to have my ideas acknowledged as mine, free-info-sharing indeed. I think within minutes they are likely to spread through the internet no-one could claim them as their own anyway perhaps even me!!

1. I have a lot of respect for KIWIS and OZ too. Your major city looks almost identical to one of our large cities here in USA. I know our “entities” really appreciate what your ANZACs have done for US over time. And the the OZ helping us SEPPOS with Pine Gap and such. I have a few friends in OZ. Fair dinkum blokes too. I’d love to come out there someday.

Regarding sharing: HaD blokes don’t seem to mind sharing even without I.P. protection. Just want to see what’s possible without outside interference. I personally get a little ho-hum with all of the FRUIT references here but I take it as a learning phase for me. I like using PC’s and laptops rather than discreet micros. But I guess you can’t get a virus or TH on a Arduino huh? Maybe not yet.

I would love to see more old-school inventions here at HaD. I like programming projects too. If you do decide to publish here on HaD please consider the lowest common denominator (translate to dummy-ese) as this is a diverse group. You have super high IQ Hawkin-types down to knuckle-draggers like Donald Trump-types (/humor!!!) – “That’s right folks… HackaDay is huuuuggggeee!” – Donald Trump

Sorry HaD folks – I don’t really believe anyone here is a knuckle-dragger like Trump. I know the average HaD’er is very smart too.. Just trying to be a funny man!

13. When you read “tinfoil-hat” stuff (/sarc) like this excerpt below from a credible paper written in 1983 related to this thread’s subject matter, you wonder how much “tin-foil” is actually intertwined in this obvious real conspiracy theory. It’s obvious this stuff exposed by Elliot above was known by “others” – somewhere in world. Just not in the modern western culture due to outside governmental interference in public education by those who falsely think suppression of information is in the interest of nation security. (Not just USA either).

…The NSA has funded four grants, while the NSF has experienced a ten percent increase in cryptology proposals submitted, the same growth it has had in other areas of theoretical computer science. But two years is a short time to measure change in a research community, and it is probably too soon to tell if the NSA restraints will have a chilling eﬂ’ect on research in public sector cryptography.

NSA actions relate directly to basic research in mathematics and computer science. For example, the security of the Rivest, Shamir and Adleman (i.e. RSA) system relies on factoring being hard. Does the NSA propose to suppress investigations on factoring integers…?

Source: Notices of the American Mathematical Society, Volume 30, Number 1, January 1983 “Primes, Codes and the National Security Agency” – by Susan Landau

Susan Landau is an American mathematician and engineer, and Professor of Social Science and Policy Studies at Worcester Polytechnic Institute. I can’t quite see her tin-foil hat that the NSA/DoD sock-puppets see so easily…

I personally “stumbled” on factoring integers for cryptography by dabbling with it during the 1980’s on DEC VAX machines. It is not in MSA (main stream academia). And all I get from HaD math experts: “stop re-inventing the wheel”. What wheel? :-P

1. ajlitt says:

“Even spookier? The finding holds up in many different bases. It was actually first noticed in base-three.”

1. Dan#734348956945785 says:

Can you prove that it holds up in all bases that are also primes though?

2. Erik Johnson – What does it matter what base it is? A prime number is a prime number no matter what base it is. In Base-10, 37 is a prime number but 37 is 25 in Hex (Base-16). 25-Hex doesn’t look like a prime but it is because 25-Hex is 37 (Base-10). So whatever base you use it still can be converted to Base-10 which is the most standard base for most “normal” humans.

37 Base-10 is 1101 Base-3 (ternary) is still a prime number as primes are ASSUMED to be base-10 or (denary or decimal). I assume that was because ancients had 10-fingers and toes (mostly). So they favored decimal as the other bases were not used until the modern age (except for Base 20 and 60 used by ancient cultures and even today).

1. I was beginning to comment on their patterns in base10 representation, not if its a prime or not.

1. Sorry Erik I was kinda’ sorta’ aiming at Dan#734348956945785 by way of your comment. HEY guess what? Dan’s enigmatic moniker number is NOT a PRIME without even checking it. Guess why? And Dan it’s a prime in any base because it can be converted to base 10.

14. RÖB says:

2 x 2 – 1 = 3
2 x 3 – 1 = 5
2 x 2 x 2 – 1 = 7
2 x 2 x 3 – 1 = 11
2 x 7 – 1 = 13
2 x 3 x 3 – 1 = 17

random ????

it’s just a spawn number

15. Lyall Burne says:

Sonofthunderboanerges thanks for what you said about Kiwis, cheers but we don’t take kindly to being compared to
Aussies. Sort of a love/hate relationship if you know what I mean. LOL.

Anyway, back to prime numbers. You requested I “translate to dummy-ese”, let me assure you my system is so simple anyone whose IQ is higher than the room temperature can easily understand it. The system can be explored further so no doubt mathematicians will property analyse it. I welcome this. Let me make it quite clear, I’m not a mathematician. Just a working man with weird, (odd), interests like prime numbers etc.

1. Oy Crickey! I’m sorry to hear that! I thought KIWIS loved my boy Mr. Hogan (with all of his American and Oz tax problems). At least you might be inclined to bend elbow with a fosters with an oz? Well anyway we have your boy Dwayne Johnson (The Rock) over here in Seppo-Land (USA). I know he was born in USA, but he did a spell in Westmere Grey Lynn with his folks who still live there with you. I love his movies. I wish Hogan would jump back in. I guess he’s really pissed with our Seppos (US-IRS) sweating him for back-taxes huh?

Fair enuf’ Lyall… some of us knuckle-draggers will appreciate the dumbing down part. Only eggheads appreciate platitudes and lofty technical jargon with misplaced loan-words and misnomers like “numbers repelling” and other anthropomorphic jibber-jabber. But I were you I would make TWO versions of the paper. One for us and one for them in their favorite highly technical convoluted jargon. But I must warn you there be monster crayfish in these waters that will try and drag you back down into the basket by severely criticizing your theories, semantics, spelling, formulas, etc.That’s just us HaD’ers. Some of us like being egotistical jerks or cyber-crayfish.Instead of constructive criticism or debate they’ll just hurl invectives and pejorative words at you. Grew a thick skin before you post it here. And don’t let it chill you either.

Some are actually Internet sockpuppets who just want to suppress any exploration into this math-technology for reasons only they and their GSA-paycheck handlers understand. Just remember what Dr. Susan Landau said up there in one of my posts in 1983. Don’t expect accolades. You may get them if your lucky. The sockpuppets will just try and tear your paper to pieces with little or no plausible rebuttals only supposition and poorly researched conjecture. That’s just my opinion. I like swimming with crayfish! :-)

16. dave says:

Perhaps this unexplained prime behavior it is a function of a universal property working in dimensions which are closed to us, but which will eventually be able to be modelled in human understandable math, like why is C finite, π, or effing magnets. Though it is a bit spooky that there might be a type of math so exotic we in our philosophers cave are simply unable to wrap our heads around it.

1. dave – here’s something in nature you probably did not know about prime numbers. You know that annoying bug that makes all that darn noise up in the trees during July and August (summer in Northern hemisphere). I imagine for Kiwis and Oz’s it is reversed. They are called CICADAS. They actually stay totally underground for several long years but only emerge to mate and die in PRIME number amount of years. Scientists are baffled by this natural anomaly. They have a working theory but it is flimsy.

Since when is C finite? There are FTL things in the Universe (i.e. the Crab pulsar, etc.). Pi is not just 3.14. The decimal fraction is as of 2015, over 13.3 trillion digits, maybe more? I must admit Feynman did a lousy job of explaining magnets. https://youtu.be/MO0r930Sn_8

1. h3ll0_w0rld says:

I love the info about Cicadas, very interesting. As far as faster than light and the Crab pulsar, erm..what?

1. h3ll0_w0rld – Nine pulsars (including Crab) have been observed exhibiting FTL (about 6X) in angular momentum. Now that’s not straight line FTL which would violate Einstein-Rosen which we all know would be sacrilegious as they were infallible (/sarc). SOURCE: (2010) Dr. John Singleton – Los Alamos National Laboratory

2. h3ll0_w0rld says:

I like where you are going but by studying gas diffusion physicists can show that while extra dimensions may apply in some situations our physical reality is bounded by the most familiar “3 physical” dimensions.

17. Emma says:

The square root of 69 = ‘ate’ something.
You might want to PRIME yourself before sinking your head into this one.

18. I still don’t think prime numbers are very strange.

Now people who give a shit about prime numbers on the other hand…

1. The security of your nation and your bank account is based on them. Elliot is correct there are some peculiarities with them. The only reason they are perceived as that as we humans are really not as advanced as we think we are. We are just scratching the surface of an iceberg of knowledge and don’t have enough time to explore it as we only have tiny lifespans compared to trees and tortoises who live a long time.

19. h3ll0_w0rld says:

If you think this is interesting you should check out Benfords law. Basically it shows that small numbers like 1 and 2 are far more common digits that later ones like 8 and 9. They use it in accounting and in court cases to show if someone is making up values for numbers instead of them being organic. Really cool stuff.

20. Jedi says:

Are primes still primes in all other systems of numeration that aren’t base 10?

1. Yes. Because PRIME numbers are “assumed” to be Base-10 ever since Euclid (or Egypt). But any base can be converted back to Base-10. Just because 25 (Base-16 or Hex) looks like a NON-PRIME Base-10 (Dec) number (25 base-10) it is not. 25 HEX is 37 DEC which is a prime number.

2. Just think of it in base 1 . physical objects like marbles. Pick a value in any base. Can you divvy up the marbles into equal groups?

21. HackJack says:

I found it hard to believe that no one, in the history of man kind, even after computers are invented, had found the properties mentioned in the article. We are talking about prime numbers. One can calculate those numbers with a few lines of code and you don’t need any floating point processor / GPU, network.

1. RÖB says:

Me to, I did this in the 80’s. Prime numbers have a definition and that is (by definition lol) is the opposite of random.

A prime number is divisible by one and itself.

So … if we say … prime number are not the numbers that are not prime then we have –

Prime numbers are not numbers that are divisible by numbers other than one and itself.

Then you realize that you are only excluding two factors from all numbers

every number can be expressed as factors and these factors have a specific and predictable sequence

1
2
3
2 x 2
5
2 x 3
7
2 x 2 x 2 x 2
3 x 3
2 x 5
11

Now if we take out the primes we have
2 x 2
2 x 3
2 x 2 x 2 x 2
3 x 3
2 x 5

and all the primes are one less than the number created with each new factor except where it has previously been factored.

22. dennisdebel says:

Its maths, of course its not random…welcome to the world of exact sciences :/

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