

Prime Explorer - MrBra
http://primes.io/

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MrBra
What are those coefficients and how are they calculated?

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jessup
Reminds me of Mills' Theorem
[http://mathworld.wolfram.com/MillsConstant.html](http://mathworld.wolfram.com/MillsConstant.html)

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radiodario
This is awesome! thanks!

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TheLarch
I spent countless hours on this with a pencil and calculator in the 90's,
laboriously back-fitting equations via recursion. This is the fulfilment of a
dream!

Also: 4x² + 570x + 20233 is 100% smooth (in the given range).

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mrcactu5
this app is using a lot of computing power, as evidence by my laptop fan
whirring.

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MrBra
Well, that's for a good cause.

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Rinum
This got me 100% prime

-622x² + 1868x + 2011

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kefka
For the range they display, that is true.

If X = 2011, then you end up with

-2511684703, which is a factor of 2011 and (-1247104+1868+1).

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orf
I'm a complete novice when it comes to maths, but I have read that primes
don't have any predictable order. Many better minds than me have proved this
true I'm guessing, but visualizations like this seem to show at least a
semblance of one. Or am I mistaken? Maybe that's what makes primes so
intriguing.

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jowiar
A lot of the pattern comes down to primes being relatively prime to smaller
numbers, which means, for example, they're 1 modulo 2, 1 or 2 modulo 3, etc.
You combine these together and find that, for example, they're all (aside from
2 or 3) going to be 1 or 5 modulo 6. Which would mean that if you lined all
the numbers up in rows of 6, all primes (aside from 2 or 3) would be in the
first or fifth columns.

So properties can give them a "shape", but they won't identify them.

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headShrinker
I wrote a very simple tool to visualize primes in the manor you describe;
shifting columns.

[http://www.795028841.com/](http://www.795028841.com/)

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MrBra
Nice! Great little tool :) Could you explain a bit better the coloring legend?
Thanks!

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headShrinker
Sure! You can type any number up to 10,000,000,000 and it will show up to
10,000 consecutive numbers.

Color key:

Prime or 1 = Purple

Divisible by 2 = pink

Divisible by 3 = green

Divisible by 5 = blue

Divisible by 7 = yellow

Composite number = gray

Clicking any color will other than prime will turn any numbers with that
factor gray.

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MrBra
Thanks. Is the choice of using different colors only up to 7 an arbitrary
choice or is the result of some particular property/theory?

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headShrinker
Good question. The first of the odd primes up to 7 occur as the most frequent
factors, other than 2. So 3,5,7 are most often reoccurring. Because of the
patterns they illustrate especially when modulating the columns it seemed to
make sense.

While all of that is true, the original concept for the project was based off
the Sieve of Eratosthenes.

[http://simple.wikipedia.org/wiki/Eratosthenes#Other_discover...](http://simple.wikipedia.org/wiki/Eratosthenes#Other_discoveries_and_inventions)

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MrBra
Hehe I asked because time ago I implemented something similar (using console
though) and by instinct I was doing the same, coloring numbers based on
divisors to see patterns, and primes, but just when I wanted to get some
conclusion out that, I had to stop and do something else, so it kinda remained
in my brain as something I had to finish to consciously understand.

Thanks for explaining then, so what we did is a Sieve of Eratosthenes... But I
have to admit that I also coded it to see which particular patterns (if any)
primes were drawing on screen.. :)

