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Images Created from Prime Numbers

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Re: Images Created from Prime Numbers

#101

Post by javajosh »

skissane wrote: post #97
javajosh wrote: post #88

Earlier quoted context omitted.

That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) In fact it seems like there must be many more non-primes than primes over any finite interval (for the simple reason that multiplication gets a lot of whacks at the piñata, e.g. producing a product landing in that finite interval.) So to answer your implied question ("why bother commucating with primes anwyay?") I guess the value of…

> That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) Any finite message (containing a finite number of symbols drawn from a finite alphabet) can be represented as a positive integer. Any positive integer n can be encoded in a prime, even a composite integer. Use the following encoding system: write 1 n times, followed by a zero, followed by an arbitrary bit string. For example, for…

I meant that the entire message is "4", not just part of the message. Obviously we can pick a subset of digits of any large prime and find anything we want. However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small.

The only reason to limit ourselves to primes is because its trivial to produce a meaningful composite, and very difficult to produce a meaningful prime. Take this message; if I was to associate it with a number (say by joining its characters as 7-bit ASCII), my money is that it's composite. (Not sure what the odds are, but I'd take 100:1 odds).

EDIT: period is `01110`, which is even, so I win.

Re: Images Created from Prime Numbers

#102

Post by jerrytsai »

JKCalhoun wrote: post #44
egypturnash wrote: post #33

Mostly unrelated but this made me check to see if the Prime Number Shitting Bear is still online. It is! https://alpha61.com/primenumbershittingbear/

The classic prime joke (saw it originally in Sci Am back in the ... 70's? 80's?). Found an approximation when I googled it: A mathematician, physicist, and engineer are taking a math test. One question asks "Are all odd numbers prime?" The mathematician thinks, "3 is prime, 5 is prime, 7 is prime, 9 is not prime -- nope, not all odd numbers are prime." The physicist thinks, "3 is prime, 5 is prime, 7 is prime, 9 ...…

Help with the joke? I understood the mathematician and the physicist, but not the engineer. What stereotype is being invoked?

Re: Images Created from Prime Numbers

#103

Post by andreareina »

criddell wrote: post #100
snowwrestler wrote: post #18

There’s a great scene in the end of the book “Contact” by Carl Sagan where the main character uses super powerful computers to find a raster image of a circle hidden deep in the digits of pi. This is supposed to be a moment of deep spiritual significance. But in fact, it’s basically an expected outcome given what we think we know about pi. At first glance these images were startling to me; that such recognizable imag…

> with an infinite numbers of primes available to us, nothing is hidden Just because there's an infinite number of primes, it doesn't necessarily mean that everything is in there, does it? For example, would you expect to find Romeo and Juliet in pi? It might be there, but I don't think it has to be there, does it?

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005). [1]

If pi is normal (as it is conjectured to be), then yeah it has to.

[1] https://en.wikipedia.org/wiki/Normal_number#Properties

Re: Images Created from Prime Numbers

#104

Post by war1025 »

oefrha wrote: post #37
war1025 wrote: post #36

Earlier quoted context omitted.

Neat. Thanks for tying it in to something real. Makes sense now that you phrase it that way. Not something I've ever looked into much.

Actually I replied without thinking too much, sorry... Your question is (probably a fair bit) stronger than the one I related to. Edited.

To take it a little further, what I was even more specifically interested in was a sequence of primes following the principle of a gray code [1] where each subsequent prime would differ from the previous prime by precisely one bit.

So basically, what is the longest sequence of primes you could find following that pattern?

[1] https://en.wikipedia.org/wiki/Gray_code

Re: Images Created from Prime Numbers

#105

Post by criddell »

andreareina wrote: post #103
criddell wrote: post #100

Earlier quoted context omitted.

> with an infinite numbers of primes available to us, nothing is hidden Just because there's an infinite number of primes, it doesn't necessarily mean that everything is in there, does it? For example, would you expect to find Romeo and Juliet in pi? It might be there, but I don't think it has to be there, does it?

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodc…

This is amazing!

So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy!

Do you know if k has to be finite?

Re: Images Created from Prime Numbers

#106

Post by skissane »

javajosh wrote: post #101
skissane wrote: post #97

Earlier quoted context omitted.

> That cannot be true. It does not contain the message '4'. (Or any of the non-primes.) Any finite message (containing a finite number of symbols drawn from a finite alphabet) can be represented as a positive integer. Any positive integer n can be encoded in a prime, even a composite integer. Use the following encoding system: write 1 n times, followed by a zero, followed by an arbitrary bit string. For example, for…

I meant that the entire message is "4", not just part of the message. Obviously we can pick a subset of digits of any large prime and find anything we want. However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small. The only reason to limit ourselves to primes is because its trivial to produce a meaningful composite, and very difficult to pr…

> However, the odds of a very large prime (like a gigabyte) looking intentionally ordered over its entire length seems vanishingly small.

Can we estimate how many gigabyte sized primes exist? I think one can do that using the prime number theorem. Let’s call that N. What is the probability one of those N messages is meaningful? I think it is a lot higher than you think it is. I think N is a very big number, even though it is very small compared to the number of composite gigabyte-sized numbers, it is still unimaginably large in absolute terms.

Re: Images Created from Prime Numbers

#107

Post by nkrisc »

javajosh wrote: post #96
nkrisc wrote: post #91

Earlier quoted context omitted.

> Why not have a SETI-in-the-primes project, after all? Sure, it can run alongside SETI-in-the-burn-marks-on-toast and SETI-the-flour-spilled-on-the-table projects.

Ah, but that's not quite the same. A message encoded in number theory itself is a far more difficult to author than just adjusting the position of 10^23 atoms.

Then it shouldn't take nearly as long!

Re: Images Created from Prime Numbers

#108

Post by SamBam »

jameshart wrote: post #94
jedimastert wrote: post #92

Earlier quoted context omitted.

This is why I love the idea of the "Library of Babel". The basic idea is that it's a vast "library" of every permutation of letters and punctuation in a book. Almost every book would be complete gibberish, but there much also be every book ever written, already created and waiting to be discovered. Is any idea truly created, or simply discovered first? Or is the simple fact of having too many possibilities man that t…

well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite. Also an infinite number of base64-encoded FLACs of the original cast recording of a rock opera based on each of those proofs, performed by David Bowie and Justin Bieber. Obviously, an infinite number of them contain logical flaws. An infinite number of them are surprisingly listena…

> well, among other things that library contains an infinite number of proofs that P=NP. Also an infinite number of proofs of the opposite.

Of course, given our current understanding of math and logic, although there would be an infinite number of proofs both for and against P=NP, only ones on one side of the argument could ever be correct, and all the others must contain logical errors, however subtle.

Assuming that the proof of P=NP isn't somehow outside of human comprehension, it means that, over an infinite period of time, a human could go over all those infinite proofs and still pick out the correct answer.

Re: Images Created from Prime Numbers

#109

Post by jonathanstrange »

andreareina wrote: post #103
criddell wrote: post #100

Earlier quoted context omitted.

> with an infinite numbers of primes available to us, nothing is hidden Just because there's an infinite number of primes, it doesn't necessarily mean that everything is in there, does it? For example, would you expect to find Romeo and Juliet in pi? It might be there, but I don't think it has to be there, does it?

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodc…

Conjectured, but still not proved to be normal. Or did I miss something this big?

Re: Images Created from Prime Numbers

#110

Post by SamBam »

criddell wrote: post #105
andreareina wrote: post #103

Earlier quoted context omitted.

> A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length k is a substring of length k appearing at a position in the sequence that is a multiple of k: e.g. the first length-k block in S is S[1..k], the second length-k block is S[k+1..2k], etc.) This was implicit in the work of Ziv and Lempel (1978) and made explicit in the work of Bourke, Hitchcock, and Vinodc…

This is amazing! So somewhere pi is a computer program that when run would set up a server with an entry for every person that has ever lived or ever will live with complete information on that person. You could theoretically see a real-time movie of anybody from life until death. Crazy! Do you know if k has to be finite?

K would have to be finite, yes, or there could never be another string after it. But k can be arbitrarily large.

Yes, Normal Numbers are awesome to think about when you first learn them. Ever tiny detail of everything that has ever happened in the history of the world, written in incredibly beautiful prose that would bring the entire planet to its knees on reading it, is somewhere in the digits of pi.

Of course, so are a lot of lies -- every possible variant of lie. And lots and lots and lots of junk. Indeed, you can make a program to search pi for any string of letters, and if your program is powerful enough you may find a 10 or 11 digit sequence of your choosing, but no more than that. And of course, searching for the string you pre-chose seems a lot more underwhelming than stumbling across the story of your life.

https://www.angio.net/pi/

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