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Prime numbers are not “nearly as scattershot” as previously thought

Surprising hidden order unites prime numbers and crystal-like materials
Patterns in prime numbers are similar to those in the positions of atoms inside certain crystal-like materials/Kyle McKernan, Princeton U

From researchers at Princeton:

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.

The researchers found a surprising similarity between the sequence of primes over long stretches of the number line and the pattern that results from shining X-rays on a material to reveal the inner arrangement of its atoms. The analysis could lead to predicting primes with high accuracy, said the researchers. The study was published Sept. 5 in the Journal of Statistical Mechanics: Theory and Experiment.

“There is much more order in prime numbers than ever previously discovered,” said Salvatore Torquato, Princeton’s Lewis Bernard Professor of Natural Sciences, professor of chemistry and the Princeton Institute for the Science and Technology of Materials. “We showed that the primes behave almost like a crystal or, more precisely, similar to a crystal-like material called a ‘quasicrystal.'”

Primes are numbers that can only be divided by 1 and themselves. Very large primes are the building blocks of many cryptography systems. Primes appear to be sprinkled randomly along the number line, although mathematicians have discerned some order. The first few primes are 2, 3, 5, 7 and 11, becoming more sporadic higher in the number line.

Torquato and his colleagues have found that that, when considered over large swaths of the number line, prime numbers are more ordered than previously believed, falling within the class of patterns known as “hyperuniformity.”

Hyperuniform materials have special order at large distances and include crystals, quasicrystals and special disordered systems. Hyperuniformity is found in the arrangement of cone cells in bird eyes, in certain rare meteorites, and in the large-scale structure of the universe.

The team showed that the order they found in the prime numbers maps to the pattern that results when X-rays interact with certain forms of matter. As a chemist, Torquato is familiar with X-ray crystallography, shining X-rays through a crystal’s three-dimensional atomic lattice. With diamonds or other crystals, this will result in a predictable pattern of bright spots or peaks, known as Bragg peaks. Kevin Mcelwee, “Surprising hidden order unites prime numbers and crystal-like materials” at Phys.org

That’s funny when you consider that the universe is supposed to be without meaning or purpose.

Hat tip: Philip Cunningham

See also: Mathematics: “Particle collisions are somehow linked to mathematical ‘motives.’”


Universe fine-tuned for carbon, hence life, says astronomer Hugh Ross at Salvo: Fine-tuning the universe for the existence of carbon is not sufficient; unless the quantity of carbon in and on a planet is also fine-tuned, physical life—and certainly advanced physical life—will not be possible.

It's not in the article. By "creator", I mean the creator of the virtual reality we find ourselves in. Not the creator of the article. mike1962
e^(i•pi) = -1 is Euler's Identity, and follows immediately from the fact that e(ix) = cos x + i • sin x. I don't see where that shows up in this article, though. jdk
Oh well, no decent font support. P.S. What's wrong with the editor, or is it just my browser? mike1962
The creator had me at ei? = -1 Ignore previous mike1962
The creator had me at ?????? = -1 mike1962
>But why the match? I would guess the match is just due to the excellent (and miraculous) way that numbers and physical reality match up. Like the Fibonnaci sequence and how sunflower seeds correspondingly spiral around in the center of a sunflower. These guys have found another such matchup between nature and math. EDTA
But why the match? bornagain77
Or maybe they are as scattershot as number theorists have thought for some time now. A quote: "The formula for the Bragg peaks was mathematically equivalent to the Hardy-Littlewood k-tuple conjecture, a powerful statement made by the English mathematicians Godfrey Hardy and John Littlewood in 1923 about which 'constellations' of primes can exist." So it looks like they just found a novel way to shine up (get it?) patterns that had already been found by other means. EDTA
Beautifully worded, Tom. OldArmy94
RJ Sawyer,its not just one "coinsidense" in in the relationships between information, mathematics, and the physical world that suggest design, it's the fact that every where we look we find very real and precise order that is practically screaming order and purpose that we logically derive from our everyday understanding of the difference between something random and something designed for a purpose. Indeed it is thousands of things from the moon's perfect position in size when compared to the Sun for a complete solar eclipse to the resonance required in making the carbon atom produced in seemingly perfect amounts, 2 life's use of immaterial information encoded in decoded to manufacture the basic machines with inside the cell. Science begins with our rational intuition, and in the design argument it is the totality of a supposed coincidences that we compare in relationship 2 our everyday experience that screams design. By the way the quasicrystal is used in one of the more popular quantum gravity proposals starting with the E8 Crystal which includes the golden ratio and units of consciousness to reconcile relativity with quantum mechanics. This idea that a biologist has to continually remind himself that what he is looking at is not designed but instead, contrary to all the evidence and probability against it, simply a crapshoot. In so many ways this is anti-science as there is no reason to assume such an illogical starting point simply to maintain a worldview where physical stuff, that is in fact simply waves of probability is primary, when in fact it seems that Consciousness and mind are primary and space time and matter are derivative. The idea we can avoid these obvious indicators of design bye suggesting our own Consciousness is an illusion and then we are one of in nearly infinite amount of universes, is a mark of desperation to keep a worldview alive rather than a true desire to understand reality. Tom Robbins
EDTA et al. - there is also a preprint on arXiv. You're right, they do just show that the density has some structure. ba77 - thanks for the link, it's nice to see the context of the work. Bob O'H
EDTA, here is an article, from May, that gives a little more background. (spoiler alert, Number theorists remain unimpressed)
A Chemist Shines Light on a Surprising Prime Number Pattern - May 2018 Excerpt: the esteemed number theorist Andrew Granville of the University of Montreal called Torquato and company’s work “pretentious” and “just a regurgitation of known ideas.” Torquato isn’t especially concerned about how his work will be perceived by number theorists. He has found a way to glimpse the pattern of the primes. “I actually think it’s stunning,” he said. “It’s a shock.” https://www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/
re 6: I suspect the latter. An exact correspondence would be massive news. jdk
It's paywalled, so I can't tell whether they have actually found an exact correspondence between primes and crystal x-ray scattering behavior, or if it's merely that the densities show a similar pattern averaged over large spans. EDTA
That unexpected mathematical pattern found in a 'quasicrystal" reminds me of these unexpected mathematical patterns found in nature:
Nature by Numbers - Fibonacci - video http://vimeo.com/9953368
Of related note:
What's the Most Efficient Design For a Solar Collector? Shape It Like a Sunflower - By Rebecca Boyle January 13, 2012 Excerpt: Concentrated solar plants use an array of mirrors, called heliostats, that move with the sun as it tracks across the sky. They are installed in concentric circles and direct sunlight to a central tower, where heat is converted into electricity. There are only a few operational CSP plants in the world, partly because they require lots of space. Each heliostat must be arranged so it faces the central tower but also the sun, all without blocking another heliostat's face. Current designs stagger the heliostats so that every other circle aligns, much like rows in a movie theater, according to MIT News. Researchers at MIT and RWTH Aachen University in Germany came up with a better way to arrange them. First they studied efficiency problems in the existing layout, finding each mirror experiences some shading and blocking problems every day. Alexander Mitsos and Corey Noone of MIT developed a computer model to bring the heliostats closer together, and they noticed the suggested patterns looked a lot like spirals found in nature. So they turned to sunflowers for further inspiration.,,, The disc florets are arranged in a spiral pattern that orients them at 137 degrees (the golden angle) with respect to each other. The angle produces a pattern of interconnected spirals that follow the Fibonacci sequence. https://www.popsci.com/technology/article/2012-01/sunflower-design-inspires-more-efficient-solar-power-plants
A further note on finding math in weird places:
The Strange Number 0.577 Keeps Showing Up in Everything And no one knows why. Excerpt: But in spite of its unwieldy nature, pi has earned its fame by popping up everywhere, in both nature and maths - and even in places that have no clear connection to the circles. And it's not the only number that has a rather eerie ubiquitiousness - for some reason, 0.577 keeps cropping up everywhere too. Known as Euler's constant, or the Euler-Mascheroni constant, this number is defined as the limiting difference between two classic mathematical sequences: the natural logarithm and the harmonic series. The harmonic series is a very famous series of numbers that you get if you start adding up numbers like this: 1 + 1/2 + 1/3 +1/4. Continue that to infinity, and you've mastered the harmonic series. The natural logarithm is far more complicated to explain than that, but the tl:dr version is if you take the difference between the values of natural logarithm and the harmonic series, you'll end up with a finite number called Euler's constant, which is 0.577 to three decimal places (and just like pi, it can go on into many, many decimals - around 100 billion).,,, ,,, Euler's constant isn't just involved in explaining that seemingly paradoxical riddle. It shows up in all kinds of problems in physics, including several quantum mechanics equations. It's even in the equations used to find the Higgs boson. And no one has any clue why. https://www.sciencealert.com/the-strange-number-0-577-keeps-showing-up-in-everything
Thank you for such a delicious turn of phrase.
I thought it would amuse people. :) R J Sawyer
does not exactly mean that there is meaning
R J Sawyer, Thank you for such a delicious turn of phrase. Andrew asauber
R J Sawyer:
The fact that there are patterns to prime numbers does not exactly mean that there is meaning and purpose in these patterns.
Only to those firmly wed to materialism ET
That’s funny when you consider that the universe is supposed to be without meaning or purpose.
The fact that there are patterns to prime numbers does not exactly mean that there is meaning and purpose in these patterns. R J Sawyer

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