[HN Gopher] Mathematicians discover prime number pattern in frac...
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Mathematicians discover prime number pattern in fractal chaos
Author : baruchel
Score : 155 points
Date : 2025-10-06 12:02 UTC (2 days ago)
(HTM) web link (www.scientificamerican.com)
(TXT) w3m dump (www.scientificamerican.com)
| baruchel wrote:
| Without paywall:
| https://www.removepaywall.com/search?url=https://www.scienti...
| _ache_ wrote:
| The conference in question:
| https://www.youtube.com/watch?v=suFspoY3bBU
|
| The article is a good "first introduction" to the presentation.
| teekert wrote:
| That image at the top is eerily like Romanesco. I actually
| thought it was at first, but it's synthetic if you look at the
| left part (... or is it?).
|
| [0]
| https://duckduckgo.com/?q=Romanesco&iar=images&t=ffab&iai=ht...
| danwills wrote:
| I think that actually is a photo of Romanesco or at least a
| pretty good fake!
| ofalkaed wrote:
| The photo's attribution gives you all you need to know.
|
| https://www.gettyimages.com/detail/photo/romanesco-broccoli-...
| teekert wrote:
| Since the left is so distant, I couldn't believe it was real.
| But it seems to be indeed. Very nice image.
| Sharlin wrote:
| I don't think it's distant? The buds are just physically
| smaller there.
| p0w3n3d wrote:
| the fractal broccoli blows my mind.
| JKCalhoun wrote:
| _Fibonacci_ broccoli.
| RansomStark wrote:
| Doodling in math class (vi hart):
|
| https://m.youtube.com/watch?v=bY1sOzTLrQQ
| YcYc10 wrote:
| It is broccoli.
| teekert wrote:
| Yeah it's also called Romanesco-Broccoli. Normal Broccoli is
| like this [0]
|
| [0] https://duckduckgo.com/?q=broccoli&t=ffab&ia=images&iax=i
| mag...
| sva_ wrote:
| Interestingly, you can often find fractal-like structures in
| nature as it is the result of maximizing surface area (to
| absorp sunlight) while minimizing space and energy used.
|
| Indeed, you can find an approximation to the (logarithmic)
| golden spiral in a romanesco, as each spiral arm is about the
| ratio of the Fibonacci sequence.
| Sharlin wrote:
| It's also one of the simplest possible shapes to encode,
| repeating the two basic instructions "grow" and "branch"
| independent of where in the plant you are.
| ndsipa_pomu wrote:
| I would have bet a sizable amount of money (maybe 50 english
| pence) that the picture was of a Romanesco. They're one of my
| favourite vegetables though not commonly found in supermarkets
| round my way.
| ReptileMan wrote:
| >In other words, just as a cloud of gas particles could be
| described deterministically if a powerful enough computer existed
|
| Let them try it with hydrogen gas.
| Quarrel wrote:
| Ok, I'll bite.
|
| Why?
|
| Isn't it still "just" a powerful enough computer?
| ReptileMan wrote:
| Hydrogen is small enough that uncertainty principle is not
| completely irrelevant.
| danwills wrote:
| I want to know more about an intuitive take on how the Zeta
| function does what it does! I know it must relate somehow to
| finding (or perhaps excluding) all the composite numbers but I'd
| really love to get more of a feeling about what each 'octave' of
| the function is adding-in. Seems like it must be something that
| 'flattens' the composites but increases sharply (in the infinite
| sum) at each prime.. but it's still a mystery to me how one could
| intuitively realise or discover that it's this specific
| function!? How did he do it?!
| sva_ wrote:
| Have you seen the 3b1b video?
|
| https://www.3blue1brown.com/lessons/zeta
| moi2388 wrote:
| See the comment from AnotherGoodName here.
| dpflan wrote:
| Is there a visualization possible of this pattern?
|
| For some reason this made me think of the Ulam Spiral --
| https://en.wikipedia.org/wiki/Ulam_spiral.
| dkural wrote:
| It is unrelated to Ulam Spiral. The patterns there are due to
| various quadratic polynomials generating a finite number of
| primes, an observation going back to at least the time of
| Euler.
| zenethian wrote:
| I wonder, has anyone tried looking for the pattern for prime
| numbers in a non-base-10 representation? I've always had a hunch
| that maybe the chaos only seems random because our representation
| of numbers could be misaligned with the pattern.
| mmargenot wrote:
| The patterns associated with primes are inherent to the numbers
| themselves and not their representations. The numbers are the
| pattern.
| psychoslave wrote:
| Yes.
|
| Also in practice we work with number representations, not
| number themselves. So there are some patterns where the
| representation is influenced by which base we encode them
| into. That's not something specific to primes of course.
|
| For example, length in term of digits or equivalently weight
| in bits will carry depending on the base, or more generally
| which encoding system is retained. Most encoding though
| require to specifically also transmit the convention at some
| point. Primes on the other hand, are supposedly already
| accessible from anywhere in the universe. Probably that's
| part of what make them so fascinating.
| bArray wrote:
| You can test it quite easily, but base doesn't seem to make a
| difference. There are some patterns that emerge when
| considering the primes spatially, though.
| alganet wrote:
| What patterns emerge when you represent primes spatially?
| kevindamm wrote:
| Elsewhere someone already mentioned the Ulam Spiral [1] and
| I'll add the Sieve of Pritchard [2] which combines the
| Sieve of Eratosthenes with Wheel Factorization. Its wiki
| page has a nice visualization. Notice that when the primes
| are arranged in a rectangular grid (rows at a time, left-
| to-right, top-to-bottom) there are entire columns that can
| immediately be eliminated from consideration.
|
| [1]: https://wikipedia.org/wiki/Ulam_spiral
|
| [2]: https://wikipedia.org/wiki/Sieve_of_Pritchard
| pcmaffey wrote:
| Try base 6.
| manwe150 wrote:
| What about base p, where each subsequent digit is the next
| prime number (also more commonly called the prime
| factorization)? Or PNS
| https://medium.com/@sumitkanoje/introducing-a-new-number-
| sys...
| 1970-01-01 wrote:
| Primes follow Benford's law. The base you choose does not
| matter.
|
| https://t5k.org/notes/faq/BenfordsLaw.html
| AnotherGoodName wrote:
| That question is literally how you derive both the zeta
| function and the prime counting functions. It also makes for an
| easy statement of why they are clearly related.
|
| It's very easy to explain too so bear with me on the following
| layman understandable explanation.
|
| First consider in base 2 every prime is of the form 2n + 1. Ie.
| every prime is odd. That's pretty understandable right? Every
| number that's not odd is a factor of 2. I could state that at
| most, above 2, only half of numbers could possibly be prime.
|
| Now lets do this with base 6 which is 2 x 3. Similarly to the
| above, in base 6 every prime is of the form 6n + 1 or 6n + 5.
| Every other form of 6n + [0,2,3,4] is going to be divisible by
| 2 or 3. This is just an extension of the above idea but we've
| done it with both 2 and 3 simultaneously. Now i can state that
| above 6 only 2/6 (1/3rd) of numbers could possibly be prime.
| Every other number is divisible by 2 or 3.
|
| Base 10 is the above idea but we do it with 2 x 5. Only 10n +
| [1,3,7,9] are not divisible by 2 or 5.
|
| Lets now continue this idea and also consider base 30. For 2 x
| 3 x 5 = 30 primes can only be of the form 30n +
| [1,7,11,13,17,19,23,29]. Any other number is a multiple of
| either 2,3 or 5. Here we see only 8/30 = 4/15ths numbers above
| 30 could possibly be prime.
|
| So... what's the formula for how many numbers can possibly be
| prime? Well if we have factors of 2,3,5... we can first work
| with the 2 and rule out 1/2 of numbers being prime (above 2
| only half of numbers can be prime). Then in the remaining 1/2
| of numbers that can still be prime, we can rule out 1/3rd of
| those numbers possibly being prime. So 1/2 x 1/3 numbers can't
| possibly be prime. Since we want to state the numbers that
| COULD possibly be prime we can state the inverse of this
| fraction. The inverse of a fraction is (1 - fraction). So (1 -
| 1/2) x (1 - 1/3) = 2/6 = 1/3. Which matches the above. Only 1/3
| of numbers above 6 can possibly be prime. Now what if we
| extended this fraction for more primes? (1 - 1/2) x (1 - 1/3) x
| (1 - 1/5) = 8/30 = 4/15 numbers above 30 could possibly be
| prime which again matches the example above. Let's continue
| (1-1/2) x (1-1/3) x (1-1/5) x (1-1/7) x (1-1/11) x (1-1/13)....
| This type of equation is known as an Euler product formula.
| This specific form which multiplys the inverse fractions of the
| primes like this is called the Reimann Zeta function. The link
| between the Reimann Zeta function and primes isn't a surprise.
| The question you asked is literally how you end up coming to
| the Reimann Zeta function -
| https://en.wikipedia.org/wiki/Riemann_zeta_function#Euler's_...
|
| Anyway the next question you may have on your mind is what does
| this series converge to? We can see as you increase the number
| of primes you get smaller and smaller fractions; 1/2 to 1/3 to
| 4/15ths of numbers possibly being prime? Well the above is how
| you derive the prime counting function;
| https://en.wikipedia.org/wiki/Prime_number_theorem#Elementar...
| and the answer is that 1/log(x) numbers are possibly prime
| above a given x.
|
| Hopefully this helps with understanding of the Riemann Zeta
| function and prime number theory in general. They are literally
| not that hard to understand in broad terms and the question you
| asked is exactly how the Zeta function came about.
| moi2388 wrote:
| Very nice explanation :)
| bobbylarrybobby wrote:
| The only thing that base affects is digital representation --
| statements like "multiples of 3 have their digits sum to a
| multiple of 3". All other behavior is a consequence of numbers'
| values, which is independent of base.
| openasocket wrote:
| There are some patterns that emerge when you look at them in
| other bases. For example, in base 6, the final "digit" (hexit?)
| is either 1 or 5, (with the exception of the primes 2 and 3).
| In base 4, the final "digit" is either 1 or 3 (with the
| exception of the prime number 2). Of course mathematicians
| generally don't talk about this in terms of their base
| representation, they usually talk about the primes modulo 6 or
| the primes modulo 4.
|
| And some of those representations actually do reveal some
| patterns. For example, an odd prime (so any prime other than 2)
| p can be written as the sum of two squares p = x^2 + y^2 if and
| only if p = 1 (mod 4). So those primes that end in 1 in the
| base 4 representation can be written as the sum of two squares,
| but the ones that end in 3 cannot. This is called Fermat's
| theorem on sums of two squares:
| https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_...
| .
|
| My guess is that there's a number of different theorems about
| prime numbers that are phrased in terms of modulo arithmetic or
| whatever that can be converted into statements about the base
| representations of primes.
|
| If I had to guess, though, I would guess there isn't a base
| where the pattern suddenly looks regular. That's very much a
| guess, but I have a couple data points to support that. The
| first is Dirichlet's prime number theorem:
| https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arith...
| . For any coprime integers a and b, the sequence a, a + b, a +
| 2 _b, a + 3_ b, ... contains infinite primes. This seems to
| imply that primes are, in some sense, evenly distributed across
| the different possible last "digits" of any base-b
| representation. There's also the Green-Tao theorem
| (https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem )
| which says that there exist arbitrarily long arithmetic
| progressions. So for any integer k, there exists some a and b
| such that a, a + b, a + 2 _b, ... a + (k-1)_ b, and a + k*b are
| all prime! I don't have a good formal argument, but that seems
| like it would introduce arbitrary "noise" into any proposed
| pattern of "digits".
|
| Finally, there's the Riemann Hypothesis. This is both my
| strongest data point and also my weakest data point. There's a
| deep relationship between the number of primes less than a
| given number and the zeroes of the Riemann Zeta function. Any
| pattern on the base-n representation of primes would imply some
| pattern on the number of primes less than a given number, which
| would in turn imply some pattern on the zeroes of the Riemann
| zeta functions. But the Riemann Hypothesis remains unsolved
| after over 150 years, despite being one of the most-studied
| problems in number theory. It seems like any pattern in the
| base-n representation would have meant some pattern in the
| zeroes of the zeta function, which means we would have made
| some progress on the Riemann Hypothesis after all this time. I
| consider this argument both very convincing and not convincing
| at all, because on one hand I'm relying on the lack of progress
| of so many people on this problem, which seems convincing, but
| also maybe it's basically just a logical fallacy, like an
| appeal to authority.
| afpx wrote:
| Not that I understand all of this. But does that mean a bijection
| exists from probability values to specific prime positions?
| dkural wrote:
| No, not at all. A bijection is a very strong requirement. There
| is no kind of morphism of any kind here, between any given
| prime and a probability, just an estimate of prime density.
| DougN7 wrote:
| Does this have any implications for cryptography where factoring
| to find large primes is involved?
| dkural wrote:
| No, it doesn't - here's an easy way to see why: You can already
| assume any conjecture to be true to see if it helps you factor
| a large number. After all it's easy to check your answer.
| profsummergig wrote:
| Looking for (possibly accidental) patterns.
|
| The obsession with prime numbers (humans decided they were
| "prime", i.e. most important, based on arbitrary considerations).
|
| It seems like a version of astrology to me.
|
| Am I wrong? I'd be happy to be proved wrong.
| eapriv wrote:
| You are wrong. Prime numbers are fundamental to mathematics.
| mjyh wrote:
| The property of certain input numbers being "prime" is required
| for RSA key generation.
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