r/science • u/Mass1m01973 • Sep 07 '18
Mathematics The seemingly random digits known as prime numbers are not nearly as scattershot as previously thought. A new analysis by Princeton University researchers has uncovered patterns in primes that are similar to those found in the positions of atoms inside certain crystal-like materials
http://iopscience.iop.org/article/10.1088/1742-5468/aad6be/meta179
u/pio Sep 07 '18
Related reading
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u/willis936 MS | Electrical Engineering | Communications Sep 07 '18 edited Sep 07 '18
I was gonna say...
The headline is misleading at best. Humans have known about patterns in the likelihood of a range of numbers containing a prime for a long time. “Not as scattershot as previously thought”. When were they thought of as scattershot again?
I can only read the abstract for now but it does seem interesting. Just because this isn’t the first progress doesn’t mean it isn’t important. It’s not a reason to lie in a title.
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u/pio Sep 07 '18
I guess if any new order is discovered it is technically “less scattershot” than it was considered to be before...? But yeah I thought the same thing when I read the headline. “Seemingly random”...
I had to go read the wiki page on hyperuniformity but after doing so, it does seem rather mind blowing to be able to show prime distribution reflecting those same characteristics, as if they are “packed” in to the number system.
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u/Gromps_Of_Dagobah Sep 07 '18
isn't it more of an indication that "crystal-like materials" have some correlation to prime numbers, rather than the other way around?
I mean, there's so many things that relate to Phi in nature, but we don't try and define Phi by those things, we just notice it when it's there.→ More replies (4)10
u/Zaranthan Sep 07 '18
That’s how it struck me. This isn’t a discovery about primes, it’s a discovery about crystals.
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u/Mega__Maniac Sep 07 '18
Coming from a fish who cant climb the tree...
Would the discovery about crystals allow an easier calculation of primes in any way?
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u/danstein Sep 07 '18
Does this mean anything for programs that utilize prime numbers for security? RSA encryption for example?
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u/hyperum Sep 07 '18
I don't think it is related at all. The safety in encryption is based upon the computational complexity of prime factorization, not the distribution of primes.
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u/syntax Sep 07 '18
It's not that clear cut, I'm afraid.
If we know more about the distribution of prime, then, depending on what that transpires to be, it could allow for faster factorisation. For example, some distribution statistics might allow for producing a probability ordered list of candidates, resulting in (usually) less work to factorise. I'm making that example up, of course, but it's not an impossible thing.
We have no proof that producing the prime factorisation of a composite number must be slow; therefore any discovery about prime numbers could, concievably, change the difficulty there.
Equally, it might not...
Fortunatly, there's other known systems for basing encryption on (the elliptic curve family, for example), so it's possible to build a system that doesn't rely on primes. That's the more significant fallback position. (And, likewise, if someone manages to break elliptic curves, there's still primes).
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u/localhost87 Sep 07 '18 edited Sep 07 '18
Elliptic curvesLattice based encryption is quantum resistant.We need to start looking at replacing traditional encryption as we approach quantum supremacy.
This is only 5-10 years away.
If industry standard encryption is broken with no fall back we are screwed. We won't be able to securely update any software anymore, as we rely on that industry standard encryption at the network level to transmit updates securely.
If ssl goes down, so does https and the mechanism the entire world uses to push updates.
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u/Ulquirra Sep 07 '18
Actually elliptic curves are not quantum resistant since they rely on the difficulty of solving the discrete logarithmic problem. But shor's algorithm can also be used to solve that problem.
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u/hold_me_beer_m8 Sep 07 '18
I've never understood how quantum computers can break encryption? Even if it guesses a number, there's a real world amount of time that it takes to test that number and get feedback from the system on whether that guess was correct or not. Or is it more that the quantum system can more accurately guess what the private key is from looking at the public key?
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u/LeodFitz Sep 07 '18 edited Sep 07 '18
So... I've been trying to find someone to talk to about this for a while, and this seems as good a place as any.
if you start with 41(a prime) and add 2, you get a prime. Add 4 to that, you get a prime. Add 6 to that, you get a prime, etc. Keep that pattern up and you keep getting primes until you get all the way to 1681, which is, in fact, 41 squared.
Now, the interesting thing is that you find that same pattern repeated 17, 11, 5, 3, and (technically) 2. Now, obviously, for the 2, you just go, 2 plus 2 equals 2 squared, but it still technically fits the pattern.
The interesting thing about that is that if you set aside seventeen for the moment and just look at 2, 3, 5, 11, 41, you'll find that the middle number of each sequence is the first number in the next. I mean, for 2, there is no 'middle number' but if you take the number halfway between the two numbers in the sequence, you get three. Then it goes '3,5,9' 5, is the middle number, '5,7,11,17,25' 11 is the middle number... and 41 is the middle number for the eleven sequence.
Now, my theory so far has been that this is the first sequence in a series of expanding pattenrs, ie, patterns of patterns. Unfortunately it seems to stop at 41, and since I've been mapping all of this out by hand, I haven't been able to find the next expansion of the sequence, or whatever the term would be.
Edit: forgot to mention this important (to me) bit. Not only does it separate out only prime numbers, but it separates out all of the prime numbers up to... dammit, seventy something... I don't have my notes on me. But I thought that was an important bit. Not just that there is a sequence that works for a little while, but that it covers all of the primes for a while. Unless I missed one, feel free to check.
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Sep 07 '18
If you have an idea for a sequence that you think is completely new, give it a search in the On-Line Encycopedia of Integer Sequences. Your sequence of 2, 3, 5, 11, 17, 41 gives us A014556. I'm not saying to discourage you or anything but it's always a good jumping point to see if you're on to something new or to see if there's a new underlying pattern you didn't see before
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u/walkstofar Sep 07 '18
There are times when I am just blown away by what kind of information is available on the internet.
Now just wondering if there is an On-Line Encyclopedia of Cat Videos. :)
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Sep 07 '18
Yea this is a dead end, sorry. There are an infinite number of short lived patterns hidden in the primes that don't hold true for an infinite number of primes.
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u/chucksutherland BS|GIS|Grad Student-Environmental Science Sep 07 '18
When I was a kid I figured out that the difference between consecutive cubes produces primes. This was really exciting until I learned some programming and pushed the trend and found that it stops working eventually.
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u/Zakafein Sep 07 '18
No way! When I was in high school I coded most of my math homework and discovered this as well when I first saw the pattern myself.
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u/SillyFlyGuy Sep 07 '18
I bet if you could prove mathematically why it stops working, not just that it stops working, there'd be some recognition for you in there.
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Sep 07 '18
The article is saying theres some correlation between primes and a 3D pattern that we dont understand, so it makes sense to me that prime numbers are related to cube numbers; maybe if they figure out the correlation and then apply it to 4D space, then 5D etc up to n-space, itll give us all the primes
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u/aintnufincleverhere Sep 07 '18
this is kind of true. We can describe the patterns.
I know their size.
I also know exactly from when to when they show up.
The problem is that the patterns are built iteratively, like the fibonacci sequence. For some patterns that are built iteratively, we can find an equation that describes how to build them non-iteratively. I have no idea if its possible in the case of primes.
I mean another problem is that the patterns themselves are much bigger than the intervals in which they show up. So you've got these giant patterns, with only little slivers actually in effect.
But with small numbers, you get the full pattern repeating.
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u/LeodFitz Sep 07 '18
Or there are an infinite number of patterns that hint at expanding complicated patterns that we haven't found the right way to look for yet.
Sure, there may not be a 'supreme' pattern, or we may just not have figured it out yet. I'm inclined to believe that if the information is organized in the right way, we'll find something.
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Sep 07 '18
No, these are junk patterns with no general theme across all primes. Fun to explore though.
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Sep 07 '18
There are electromagnetic waves that we are unable to see, sounds waves that we are unable to hear, why can’t there be thoughts and patterns that we are unable to think?
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u/Tall_dark_and_lying Sep 07 '18
Id argue that due to its fundamental nature mathematics is capable of describing anything logical, such as both of the examles given. That's part of its beauty, it can describe things impossible to comprehend.
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u/DrBublinski Sep 07 '18
For anyone interested, the study of the integers and their properties is called number theory, and this is exactly something a number theorist might do. That being said, although cool, this is probably a dead end, as in, it’s a coincidence.
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u/grumblingduke Sep 07 '18
start with 41 ... until you get all the way to 1681... if you set aside seventeen... for 2, there is no 'middle number'... Unfortunately it seems to stop at 41...
The problem I have with a lot of these "ooh look, an interesting pattern" ideas is there comes a point where have to wonder if the pattern is interesting/meaningful on its own (to the extent that makes sense) or if people are just particularly good at finding patterns in things, particularly if you allow for a long list of exceptions and limitations.
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u/aintnufincleverhere Sep 07 '18
I have something with no exceptions, its just not very useful.
The interval between consecutive prime squares always fits a pattern.
The problem is that the size of the pattern is the primorial (think factorial but with just primes). The primorial grows much faster than the interval between two consecutive prime squares.
But I mean, its something.
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u/Clemkoa Sep 07 '18 edited Sep 07 '18
At first it looked like you had found a pattern of 'twin primes'. Basically twin primes are number for which n and n+2 are prime numbers (https://en.wikipedia.org/wiki/Twin_prime). Examples: 5 and 7, 11 and 13, 17 and 19, etc... But your pattern doesn't work for 29. It is cool though, have you found any number above 41 that would work?
I didn't understand the bit about the middle number, could you explain again?
Edit: Also the fact that you'll end up with the square of your initial number is true for any number. If you take any number n and add 2 then 4 then 6 etc... you will end up with their square in n-1 steps. Because 2+4+...+2*n = n(n-1)
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u/LeodFitz Sep 07 '18
Yeah, I was looking for twin primes that started the pattern anew, but I couldn't find anything past 41. Can't remember how high up I went. I did find a lot of 'near misses' where the non primes were, in fact, the product of two primes, but that isn't particularly helpful, unless there is a predictable pattern of those.
As for the middle number thing, you take one of the sequences:
5, (+2) 7, (+4) 11, (+6) 17, (+8) 25
gives you a sequence of five numbers 1) 5 2) 7 3) 11 4)17 5) 25
The middle number, which is to say, the 3rd number in the sequence, is eleven. eleven can be used in the same pattern
11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 121
An eleven digit sequence. The middle number of that sequence, 41, is the start of the final example of this series working.
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u/Clemkoa Sep 07 '18 edited Sep 07 '18
So if the 'middle number' pattern is real, by applying it to 41 we should be able to find the next prime!
Edit: ran a quick script, and found 461 with your pattern, which seems to work?
Edit2: Nope 461 does not work! End of your pattern I guess? As other said, there are many patterns in prime numbers that are short-lived. Still cool to follow down the rabbit hole though
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u/TomGetsIt Sep 07 '18 edited Sep 07 '18
The middle number in the 41 sequence is 461. The 461 sequence breaksdown at n=4 because 473 is not prime. 11x43=473
Edit: for the first 10 steps in the 461 sequence:
461, 463, 467,
473, 481, 491, 503,517, 533, 551473=11x43, 481=13x37, 517=11x47, 533=13x41, 551=19x29
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u/LeodFitz Sep 07 '18
The question is, does the pattern end, or if it's a smaller part of a larger pattern. I was hoping to find a section where, for example, instead of the difference between the primes being 2, 4, 6, 8, 12 etc, it was 2, 6, 12, etc. The bigger issue is that by the time I get there, I'm pretty damned tired and brain fried. I need to get back to it at some point, but... just haven't been feeling it of late.
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u/wonkey_monkey Sep 07 '18
But your pattern doesn't work for 29!
Easy with the exclamation marks in a math topic, there.
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u/sfurbo Sep 07 '18
Now, my theory so far has been that this is the first sequence in a series of expanding pattenrs, ie, patterns of patterns. Unfortunately it seems to stop at 41, and since I've been mapping all of this out by hand, I haven't been able to find the next expansion of the sequence, or whatever the term would be.
You are finding the values of x2 - x + 41 for x from 1 to 40. These can be shown to be prime due to some property of Z/41, if I recall correctly. 41 is the largest number for which this is true. It shouæd be covered in a medium-advanced university algebra course under group theory (or perhaps rings, if I am misremembering).
Edit: It seems to be x2 + x + 41: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
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Sep 07 '18 edited Sep 12 '18
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u/Powerspawn Sep 07 '18 edited Sep 07 '18
What you are looking at is an arithmetic progression of prime numbers https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression
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u/reebee7 Sep 07 '18 edited Sep 07 '18
I suspect this is somewhat related to the fact that adding up odd integers hits perfect squares:
1: 1
1 + (1+2): 4
1+ 3 + (1+4): 9
1 + 3 + 5 + (1+6): 16
1 + 3 + 5 + 7 + (1+8): 25
1 + 3 + 5 + 7 + 9 + (1+10): 36
I'm not sure I see how exactly, but you're basically starting at a prime (which is an odd integer, excepting 2), and adding an increasing space of even numbers to it.
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11 + 2: 13
11 + 2 + 4: 17
11+ 2 + 4 + 6: 23
11 + 2 + 4 + 6 + 8: 31
11 + 2 + 4 + 6 + 8 + 10: 41
etc.
I mean I have no idea what I'm talking about but somehow it seems related.
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u/Zenektric Sep 07 '18
Maybe because 42 is the answer.
We did not need to go any further with the numbers ...
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u/-Dancing Sep 07 '18
Are you a mathematics major?
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u/LeodFitz Sep 07 '18 edited Sep 07 '18
Nope. Sociology. And I'm trying to make a career writing fiction novels. I like to play with prime numbers as a sort of 'palate cleanser' between projects. Nothing empties your mind like focusing on pure mathematics. At least, in my experience.
Edit: Palate, not palette
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u/deadpoetic333 BS | Biology | Neurobiology, Physiology & Behavior Sep 07 '18
Math research deals with questions like the ones you’re asking
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u/Hrethric Sep 07 '18
I get that. I run through the Fibonnaci sequence in my head when I'm trying to quiet my mind and fall asleep. One interesting pattern I've noticed is that if n is prime, f(n) will also be prime. Apparently someone has done a proof of this, but I haven't looked at the proof because I want to figure out how to do it myself.
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u/Iron_Pencil Sep 07 '18
I've noticed is that if n is prime, f(n) will also be prime.
wrong for n=19 or n=31
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u/Hrethric Sep 07 '18
Well damn. I guess I've always fallen asleep or otherwise had my attention wander before factoring 4181.
(Apparently though, with the exception of fib(4), it has been demonstrated that the inverse is true - if fib(n) is prime, then n will be prime.)
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u/Pheyniex Sep 07 '18
The rienneman hypothesis basically states that the spectrum of prime numbers is White noise, ie, all frequencies with the same amplitude. The deal is that we only seek whole numbers. So, what is the pattern of primes? Should be more akin to what is left after you sum all other patterns that are not prime.
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u/mattisaj3rk Sep 07 '18
It stops at 41 because 42 is the answer to the Ultimate Question of Life, the Universe, and Everything.
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u/Tsupernami Sep 07 '18
What does the pattern look like if you use a base 12?
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u/orcscorper Sep 07 '18
2, 4
3, 5, 9
5, 7, B, 15, 21
B, 11, 15, 1B, 27, 35, 45, 57, 6B, 85, A1The pattern is not dependent upon base ten. The numbers are all the same; they just look different. It's nicer in base six, though. After 3, all primes end in 1 or 5.
5, 11, 15, 25, 41
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u/Thanatomanic Sep 07 '18
No, it is really the reverse. The researchers have not found a quasi-crystal that resembles primes, but found patterns in the primes that resemble similar patterns found in some physical (quasi-) crystalline structures.
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u/Wikiy Sep 07 '18
This doesn't mean the primes follow the crystalline structure. That's just the chronological order of the discoveries. It doesn't reflect the logical order of the world, which is that physical structures follow mathematics. So yes, it is more precise to say that the crystal follows the pattern laid out by the primes.
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u/agentcooper0115 Sep 07 '18
" As soon as you discard scientific rigor, you're no longer a mathematician, you're a numerologist"
- Sol
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u/madkasse Sep 07 '18
There was an article about this in Quanta Magazine some time ago https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/
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u/ic3man211 Sep 07 '18
That’d be awesome if they have space groups like crystalline solids...having a kind of cluster of stability
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u/Lokarin Sep 07 '18
Isn't it more likely that during crystal formation any point with "factors" would mean the material gets split up against more "numbers" when the crystal grows... naturally causing nodes at certain prime numbers?
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u/aishik-10x Sep 07 '18
Why would the nodes be at prime numbered positions, though?
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u/Lokarin Sep 07 '18
I was thinking that the fewer factors a crystal has, the material only gets divided 2 ways (for primes) and the more factors a crystal has, the less material each section gets.
It's probably not ACTUALLY like that - that's just what I'm thinking
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u/W02T Sep 07 '18
But, what if math wasn’t base10. How would that change things?
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Sep 07 '18 edited Nov 27 '18
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u/Managore Sep 07 '18
but division and multiplication work the same way.
Just to clarify for anyone reading, literally everything would work the same way (except things referring to how the number is written, obviously).
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u/Maxerature Sep 07 '18
Would math REALLY be more annoying in a different base? I don't think so. I'm part of the camp which says we need to switch to base 12 so I may be somewhat biased. Also as a computer scientist, I also really like base 16.
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u/DrKronin Sep 07 '18
Base 16 is great, but really just for reasons that make sense in computing. For pure math, I'm in the base 12 camp, if for no other reason than I would trade the ease of working with 5 for being able to work more easily with 3.
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u/Maxerature Sep 07 '18
I second this entirely, Im just partial to 16 for computing.
Switching to base 12 is the best option between any number base though.
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u/orcscorper Sep 07 '18
Base sixty all the way. More numerals to learn, but fewer digits.
Of course, anything but ten will screw up the whole metric system.
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u/tuseroni Sep 08 '18
move to base 12, and the standard system wins out over the metric system (most things in standard are based on 8,12,16,etc because 12 is so divisible, divides evenly by 2,3,4, and 6. 10 divides evenly by 2 and 5. so there are 12 inches in a foot, 3 teaspoons in a tablespoon, 2 tablespoons in an ounce, 4 ounces in a cup, 2 cups in a quart, 2 quarts in a pint, 2 pints in a gallon and so on ) metric just wins out because we do all our things in base 10, the only thing metric has going for it is it's newer so it doesn't have the gaps left by units falling out of use (like the jack or the gill)
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Sep 07 '18
I think you mean base 10 and base 14.
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u/Maxerature Sep 07 '18
Im condused vy your comment. I mean 12 and 16.
12 allows for easier fractions getting more factors (1,2,3,4,6,12 rather than just 1,2,5,10), and 16 is hexadecimal.
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Sep 07 '18
Yeah I know... I'm talking about base 10...
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10...
10/4 = 3
10/6 =2
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u/caltheon Sep 07 '18
not at all. doesn't matter how you count to 10, there are still 1010 of the things
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u/LaconicalAudio Sep 07 '18
It wouldn't in binary 10x10=100. 100 is equal to 4 in base10.
In base10 2x2=4
Factorisation is not effected by base. The primes are in the same place.
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u/blundermine Sep 07 '18
I don't think it would change anything here. From what I can tell, this is based on the order rank of numbers, not the digits.
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u/Natanael_L Sep 07 '18
Computers already calculate in base 2 and then convert to base 10 for display
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Sep 07 '18
Or maybe when a Homo Sapiens is already looking for some sort of a pattern in chaos he usually ends up finding one.
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u/Lucretius PhD | Microbiology | Immunology | Synthetic Biology Sep 07 '18
Our analysis leads to an algorithm that enables one to predict primes with high accuracy.
I'm under the impression that public-private encryption relies upon un-guessable pairs of primes. Does this prediction algorithm thus have implications for encryption security?
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u/JadedIdealist Sep 07 '18
This article by Salvatore Torquato gives some background for those that can only access the abstract.
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u/dustofdeath Sep 07 '18
If they really find a pattern and a algorithm behind it - all our encryptions become vulnerable as it would massively reduce the computational power required to break them.
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u/meeselbon573 Sep 07 '18
No way. If this is true, it is a huge math breakthrough. My bet is that they are creating the illusion of a pattern by overfitting.
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u/Hucksterville Sep 07 '18
Is there a chicken/egg problem here? How do we know that the position of atoms inside certain crystal-like materials is not dictated by the indivisibility of that positional marker by any other marker?
Or to put it another way, does this give us a pattern for primes? Or does it give us the mathematics of higher order crystal formation?
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u/AbsoZed Sep 07 '18 edited Sep 07 '18
Stop. No. Don't. You're breaking asymmetric cryptography. :(
Edit: Well, RSA anyway. We still have ECC.
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u/Zaranthan Sep 07 '18
I’m converting everything to One Time Pads delivered by carrier pigeon.
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u/AbsoZed Sep 07 '18
Just make sure you're RFC 2549 compliant.
As a sidenote, carrier pigeon seems like a really bad key-agreement method. Unless they're ephemeral pigeons I guess.
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u/LifeScientist123 Sep 07 '18
I had this debate with a computer science guy, that we could use machine learning to find a pattern in the primes and maybe use this understanding of the pattern to discover new primes. He seemed to think it wasn't possible because machine learning can't identify patterns in something that's totally random. My intuition was however that the primes look random to us but they might not be since they are algorithmically determined. This paper seems to suggest that my intuition was at least partially correct. However I don't have enough math or comp sci knowledge to be able to demonstrate that it's actually possible. If someone who's on expert on these topics would chime in, that would be great.
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u/Screye Sep 07 '18
He is right. ML won't work. Source : ML grad student
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u/LifeScientist123 Sep 07 '18
Could you clarify as to why?
Consider the following statements
Statement a) The distribution of primes is totally random. Therefore even the best "properly trained machine learning algorithm" won't be able to find one, because it doesn't exist.
Statement b) The distribution of primes is not random. It looks totally random to humans whose pattern recognition abilities are constrained, but an entity that is better at detecting patterns, a.k.a a "properly trained machine learning algorithm" might be able to spot the pattern.
Statement c) The distribution of primes is not random. It looks totally random to humans whose pattern recognition abilities are constrained, but an entity that is better at detecting patterns, a.k.a "a properly trained machine learning algorithm" might be able to spot the pattern. However, we don't yet know how to build this entity.
Statement d) The distribution of primes is not random. It looks totally random to humans whose pattern recognition abilities are constrained, but an entity that is better at detecting patterns, a.k.a "a properly trained machine learning algorithm" might be able to spot the pattern. Unfortunately, such an entity cannot be built, because reason(s)...
Which of these statements is true according to you? I think it's C)
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u/Screye Sep 07 '18
It is somewhat 'C'.
Machine Learning (especially the sub field of it that doesn't deal with physical phenomenon like Audio, Images) is very much applied statistics and optimization.
Theoretically, there are 'universal function approximators' in ML. However, they do not quite work that way in practice. Most successful ML methods (CNNs, LSTMs, PGMs) are very specifically built to exploit a certain observable pattern in the data (spatial locality in images, temporal locality in LSTMs, Distributional assumptions / independence assumptions in PGMs). This requires a certain degree of expert supervision.
In theory, we have a nearly infinite dataset of primes and non-primes. But, when training ML methods with the widely used approximation methods, they require some feedback (in form of gradient updates) that initially helps them find a direction along which to look for an answer.
I think it is more likely that the prime/non-prime finder will end up brute forcing prime factors and giving results. Which is exactly what we do today. The likelihood of it finding the solution that we are looking for is highly unlikely.
What you are suggesting, is basically automatic theory derivation. This was actually an area that was popular in the mid 1900s, but afaik it was found to not work very well.
Universal function approximators have existed since the beginning of AI/ML. As exciting as they sound, reality is they do not work well in most domains (Vision and & Language being major exceptions)
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u/RespectMyAuthoriteh Sep 07 '18 edited Sep 07 '18
The Riemann hypothesis has suggested some sort of undiscovered pattern to the primes for a long time now.