Prime After Prime
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Prime After Prime
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Re: Prime After Prime
#2A puzzling article; starts of saying there's no pattern, then shows a pattern.
In the hope that people who know more about primes that me (this will not be unlikely) are reading this:
How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them in 2d and 3d, and altering the way they're displayed (spirals, ovals, wrapping them as square spirals etc) and seeing if anything comes out. I mean, on the off chance that it might be easier to detect a pattern at a human level than apply strict mathematical techniques. Is this still an area under active research, or was it rather quickly tried and then given up with "nope, this isn't getting us anywhere"?
As you can probably tell, I'm not a mathematician. I just have a sort of superstitious belief that there's something "natural" about primes, and that there'll be a pattern there somewhere. I've read about people with mental health issues who somehow are able to tell if a number is prime and assuming that this is accurate then again, unless they are somehow able to run a learned algorithm in their heads stupidly fast they're therefore tapping into something natural. Or perhaps these people weren't assessed by anyone who actually knew what a prime number was.
Re: Prime After Prime
#3A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
That could make a pretty neat crowdsourcing project. Come up with a parameterized set of prime number visualizations. Have a website that draws ("draw" in both the sampling and graphical sense) pairs of visualizations, asking the visitor to select which is most "interesting".
Re: Prime After Prime
#4A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
When I was a kid, my dad had a postcard lying around with a reply from Isaac Asimov about a pattern my dad said he found in primes by writing them in a table or something. The answer was basically, "when you look at the page your visual cortex is doing a ton of math you're not factoring in..."
Re: Prime After Prime
#5A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
Re: Prime After Prime
#6Let p-prime. Consider 3 "probabilities": a) p+2 is prime; b) p+4 is prime; c) p+6 is prime. Obviously p(c)=2p(a)=2p(b).
Probability of p+30 being prime is even higher.
Re: Prime After Prime
#7A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
there are a number of ways to answer your question, so consider the following only one:
i do research into the underlying pattern of primes, and do so from the intended inference that there does exist an algorithm that returns primes based on configurable inputs
> starts of saying there's no pattern, then shows a pattern
there are many patterns in the prime number sequence
here is a seemingly ludicrous pattern in the sequence of primes 468,395,662,504,823 + 205,619 · 223,092,870 · n, for n = 0 to 23 (o)
but i think when people say things like: If there’s a pattern, it’s inscrutable; from the article they are referring to an ability to answer questions for all primes such as: what is the three thousandth and one prime after 17, or what prime follows and precedes 113; those kinds of questions are currently unanswerable for all numbers efficiently
> How hard have people looked for pattern visually?
the concept of visual is very subjective
numbers are abstract concepts and how you choose to visualise them is simply a matter of choosing how the abstraction you use will appear, i can make any prime number appear anywhere in a visualisation depending on how i configure the visuals
> the way they're displayed (spirals, ovals, wrapping them as square spirals etc)
dreaming up display methods, and developing them, is math
there is a famous question in mathematics called the riemann hypothesis(i)(ii) where one could argue that complex numbers become a clever abstraction for visualising of primes
> I've read about people with mental health issues who somehow are able to tell if a number is prime and assuming that this is accurate then again, unless they are somehow able to run a learned algorithm in their heads stupidly fast they're therefore tapping into something natural.
confirming if a number is prime or composite, what is called a primality test(iii), is a 'solved' problem with an efficient algorithm
(o) https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
(i) https://www.youtube.com/watch?v=d6c6uIyieoo
Re: Prime After Prime
#8Let p-prime. Consider 3 "probabilities": a) p+2 is prime; b) p+4 is prime; c) p+6 is prime. Obviously p(c)=2 p(a)=2 p(b). Probability of p+30 being prime is even higher.
I assume you mean P(c) = P(a) P(b) + P(b) P(a) (i.e., you can get to p+6 as p+2+4 or as p+4+2) = 2 P(a) P(b)? But even so, that's not true, because you're messing with conditional probabilities in a weird way.
Let k be any integer. Consider the probabilities a) k+2 is a multiple of 10, b) k+4 is a multiple of 10, and c) k+6 is a multiple of 10. P(a), P(b), and P(c) are all 0.1, but your approach would claim that P(c) is 0.02. And P(k + 30 is a multiple of 10) is also still 0.1.
Re: Prime After Prime
#9A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
There are certainly patterns. Try this:
Draw lines from the origin with slopes 1, 1/2, 1/3, 1/4, ...
A number, y=n, is prime if no line intersects y=n at an integer other than n.
You can also start out with a line of points, then repeatedly place a new line of points, translated up one unit and right one unit from the start of the last line and increase the distance between each point by one from the last line. [There are many other simple constructions and many repeating sub-units with definite patterns.]
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http://divisorplot.com records more patterns.
I currently have a demo [0] up that lets you explore these drawings in projective spaces. [Note that the default scene is showing only a scaled portion of the underlying data.]
Re: Prime After Prime
#10A puzzling article; starts of saying there's no pattern, then shows a pattern. In the hope that people who know more about primes that me (this will not be unlikely) are reading this: How hard have people looked for pattern visually? I saw a few pieces about it online but I'd have thought that it would be rather easy to take the first, say, 10 million primes and subject them to all sort of techniques displaying them…
I think the author was trying to say that there seems to be no simple pattern. The original sequences are meant to demonstrate how hard it is to see a pattern. However, a relatively recent empirical discovery found very simple statistical biases between consecutive primes. Mathematicians were shocked by these results because they were so simple. This article demonstrates and further explores this surprising phenomenon.