Prime numbers

[Mello_]

As I know prime numbers are now biggest math puzzle. Its last historical math mystery. Its Cosmos code. Cosmos language. Universities, Nasa and many offer big prizes who found right formulae. We need it for passwords.  Big high tech computers trying to find next biggest number. Yada yada...

 

Lets ignore 3 for second.

All prime numbers are 6n +-1

5=6-1

7=6+1

11=(6x2)-1

13=(6×2)+1

17=(6×3)-1

...

Base its 6. Multiply by n. + or - 1.

 

Edited April 21, 2020 by Mello_

[Golden Duck]

10 minutes ago, Mello_ said:

As I know prime numbers are now biggest math puzzle. Universities, Nasa and many offer big prizes who found right formulae. We need it for passwords.  Big high tech computers trying to find next biggest number. Yada yada.

 

Lets ignore 3 for second.

All prime numbers are 6n +-1

5=6-1

7=6+1

11=(6x2)-1

13=(6×2)+1

17=(6×3)-1

...

Base its 6. Multiply by n. + or - 1.

 

All prime number greater than 3.

[Mello_]

3 minutes ago, Golden Duck said:

All prime number greater than 3.

Yes. Take any known prime number.

6n +-1

Simple as that.

[Golden Duck]

1 minute ago, Mello_ said:

Yes. Take any known prime number.

6n +-1

Simple as that.

2 and 3 are prime numbers.

[Mello_]

Just now, Golden Duck said:

2 and 3 are prime numbers.

Ok. Im sure some mathmatician could writte down formula better than me. With sin con i derivation imaginary numbers negative numbers to get to 2 and 3.

But nevertheless. Its amazing to me that all rest prime numbers are 6n plus minus 1.

Thats pattern.

[zep73]

1 minute ago, Mello_ said:

Ok. Im sure some mathmatician could writte down formula better than me. With sin con i derivation imaginary numbers negative numbers to get to 2 and 3.

But nevertheless. Its amazing to me that all rest prime numbers are 6n plus minus 1.

Thats pattern.

You think it's about the great boat from the revolution, sorry, the great beast from the revelation?

[Mello_]

Just now, sci-nerd said:

You think it's about the great boat from the revolution, sorry, the great beast from the revelation?

Sorry? Wtf

[Golden Duck]

6 has prime factors of 2 and 3

6n is therefore divisible by 2 and 3

6n + 1 might be prime

6n + 2 and 6n + 4 are clearly divisible by 2

6n + 3 is clearly divisible by 3

6n + 5 = 6(n + 1) - 1 might be prime

We've accounted for all the natural numbers we can represent with one hand.

Edited April 21, 2020 by Golden Duck

[ChrLzs]

6 minutes ago, Golden Duck said:

6 has prime factors of 2 and 3

6n is therefore divisible by 2 and 3

6n + 1 might be prime

6n + 2 and 6n + 4 are clearly divisible by 2

6n + 3 is clearly divisible by 3

6n + 5 = 6(n + 1) - 1 might be prime

We've accounted for all the natural numbers we can represent with one hand.

Exactly - there's no magic here, it is just a consequence of simple arithmetic.  Thusly, it's also no coincidence that 6 just happens to be the product of multiplying 2 and 3, which are the two 'exceptions'....

If you don't get it, just re-read Duck's step by step analysis, until you go .. "ooooh, I get it"..... 

[Mello_]

14 minutes ago, Tuco's Gas said:

Isn't "6n-1" more elegant?

Here's some of the biggest remaining questions re primes...

https://www.businessinsider.com/cool-math-prime-numbers-2018-3

Sometimes after 6n you must add one or sub one.

[Mello_]

23 minutes ago, Golden Duck said:

6 has prime factors of 2 and 3

6n is therefore divisible by 2 and 3

6n + 1 might be prime

6n + 2 and 6n + 4 are clearly divisible by 2

6n + 3 is clearly divisible by 3

6n + 5 = 6(n + 1) - 1 might be prime

We've accounted for all the natural numbers we can represent with one hand.

You...didnt understand. Sorry.

[zep73]

25 minutes ago, Mello_ said:

Sorry? Wtf

I was making fun of those who see sixes everywhere. You know, the bible and 666.

[Mello_]

Just now, sci-nerd said:

I was making fun of those who see sixes everywhere. You know, the bible and 666.

Aww. Sorry dude. Its late here in Eu. My bad. 

[Tiggs]

Believe it’s essentially a variant of Eratosthene’s sieve. 
 

Taking 2 and 3, you get a repeating pattern 6 long

010101 OR

001001

======

011101

Where the 0’s are potential primes
 

Take the next prime (5) and you’ll get a pattern 30 digits long:

011101011101011101011101011101 OR

000010000100001000010000100001

============================

011111011101011101011101111101

30n +1 +7 +11 +13 + 17 + 19 + 23 + 29

Then you’d take the next prime (7) and get a pattern 210 long.

Believe it should accurately find all the primes up to the square of the next prime found (So with 210, accurate up to 11 squared = 121).

Then 210 * 11 = 2310, accurate to 169, etc.

 

32 minutes ago, Tuco's Gas said:

Isn't "6n-1" more elegant?

Doesn’t work for 7, 13 etc.

[Mello_]

19 minutes ago, sci-nerd said:

I was making fun of those who see sixes everywhere. You know, the bible and 666.

I will be nerdy and say its Carbon. 666. 6 electrons 6...Thats why Carbon is Carbon. Star stuff and bond of all living things.

Or fringy and mentioned Carl Jung synchronicty. As Einstein said it when you have coincidence its when God wants to stay anonymus.

Edited April 21, 2020 by Mello_

[Golden Duck]

18 minutes ago, Tiggs said:

Believe it’s essentially a variant of Eratosthene’s sieve. 
 

Taking 2 and 3, you get a repeating pattern 6 long

010101 OR

001001

======

011101

Where the 0’s are potential primes
 

Take the next prime (5) and you’ll get a pattern 30 digits long:

011101011101011101011101011101 OR

000010000100001000010000100001

============================

011111011101011101011101111101

30n +1 +7 +11 +13 + 17 + 19 + 23 + 29

Then you’d take the next prime (7) and get a pattern 210 long.

Believe it should accurately find all the primes up to the square of the next prime found (So with 210, accurate up to 11 squared = 121).

Then 210 * 13 = 2730, accurate to 169, etc.

 

Doesn’t work for 7, 13 etc.

I see it more as the first optimisation in a prime test.  This way you don't try dividing by an odd number that's divisible by three.

[Tiggs]

14 minutes ago, Golden Duck said:

I see it more as the first optimisation in a prime test.  This way you don't try dividing by an odd number that's divisible by three.

As it's a repeating pattern, that's effectively what it is. 

[Rlyeh]

10 hours ago, Mello_ said:

Base its 6. Multiply by n. + or - 1.

 

These aren't primes

6*4+1 = 25 = 5*5

6*6-1 = 35 = 5*7

6*8+1 = 49 = 7*7

[Tiggs]

2 hours ago, Rlyeh said:

These aren't primes

6*4+1 = 25 = 5*5

6*6-1 = 35 = 5*7

6*8+1 = 49 = 7*7

Yeah. There's several examples where it's not prime. But every prime number will fit that general pattern.

Think of them as potential primes, if you will.

Believe the general rule is that it's accurate until the square of the next prime in the pattern -- so if you start with 2 & 3, giving the repeating pattern 6 long -- 011101 -- then it's accurate until the square of the next prime (which is the next 0 after the first one in that pattern) -- so, the 5th digit -- and 5 * 5 = 25, which is the first non-prime, which you've discovered above.

To get the next pattern -- you multiply the current pattern length by the next prime(5) to give you a new pattern length of 30, then OR the repeating patterns together.

Like this:

011101011101011101011101011101 OR --> repeating pattern of 011101 (from the combined primes of 2 & 3)
000010000100001000010000100001    --> repeating pattern of 00001 (for the next prime, 5)
==============================
011111011101011101011101111101 --> a new repeating pattern, 30 long (1 * 2 * 3 * 5) <-- first 4 primes

Giving the general pattern
30n +1 +7 +11 +13 +17 +19 +23 +29
(From the 0's in the result above)
Which applies until the next prime (7) squared = 49.

If you wrap that pattern around, you'll note that 25 no longer shows up as a potential prime, and nor does 35, as we've just removed all the possible 5's from the repeating pattern.

You should be able to calculate every single prime number from those rules.

The main issue in doing so is storage, as the product of all the prime numbers gets pretty big, pretty fast -- so when you're 9 primes in, you get a pattern that's 9,699,690 long (as 1 * 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 = 9,699,690), but is only accurate to 23 squared (529), as 23 is the next prime.
 

5 hours ago, Tuco's Gas said:

https://www.businessinsider.com/cool-math-prime-numbers-2018-3

The infinite Twin primes conjecture is an interesting one, because if it's true, then a proof should exist using the simple 6n +/-1 pattern, above.

If true, then should be able to show that the number of back-to-back pairs created by the repeating pattern was such that it would be impossible for them all to be removed by subsequent prime dividers.

Given that the prime accuracy is ensured to the square of the next prime -- it feels like it's probably true -- but I don't have the necessary math chops to back that up, sadly.

[Rlyeh]

12 minutes ago, Tiggs said:

Yeah. There's several examples where it's not prime. But every prime number will fit that general pattern.

Think of them as potential primes, if you will.

Believe the general rule is that it's accurate until the square of the next prime in the pattern -- so if you start with 2 & 3, giving the repeating pattern 6 long -- 011101 -- then it's accurate until the square of the next prime (which is the next 0 after the first one in that pattern) -- so, the 5th digit -- and 5 * 5 = 25, which is the first non-prime, which you've discovered above.

To get the next pattern -- you multiply the current pattern length by the next prime(5) to give you a new pattern length of 30, then OR the repeating patterns together.

Also as n increases so does the frequency of composite numbers. 

I've written some code that tests their theory;

https://godbolt.org/z/TiyX-x

[Tiggs]

2 hours ago, Rlyeh said:

Also as n increases so does the frequency of composite numbers. 

I've written some code that tests their theory;

https://godbolt.org/z/TiyX-x

Coded in C++. Brave man.

I'd totally expect the composites to stack up over time, as the basic 6 digit-long pattern only excludes multiples of 2 & 3's.

As soon as you hit anything that's a product of two higher prime numbers -- then it'll give a false positive -- and you'll find more and more of those, the higher the numbers go.

At it's heart -- it's really simple.

You're just excluding all the possible multiples of two -- so 01 01 01 01 01 01 etc.

and then all multiples of 3 -- so 001 001 001 001 001 001, etc.

And when you OR those together, you get a repeating pattern, 6 long -- 011101 011101 011101

So you can describe the gaps -- the 0's -- as 6n +/- 1 (or 6n +1/+5, if you prefer).

Which are the only places a prime > 3 could possibly be (as everywhere else is divisible by either two or three).

[simplybill]

This may be off-topic, but it does have to do with prime numbers. In Dr. Oliver Sacks’ book ‘The Man Who Mistook His Wife For a Hat’, the chapter ‘The Twins’ describes two autistic brothers who played a numbers game of exchanging prime numbers. The numbers had as many as 20 digits. A quote from the essay:

”They drew apart slightly, making room for me, a new number playmate, a third in their world. Then John, who always took the lead, thought for a very long time - it must have been at least five minutes, though I dared not move, and scarcely breathed - and brought out a nine-figure number; and after a similar time his twin Michael responded with a similar one. And then I, in my turn, after a surreptitious look in my book, added my own rather dishonest contribution, a ten-figure prime I found in my book. 

There was again, and for even longer, a wondering, still silence; and then John, after a prodigious internal contemplation brought out a twelve-figure number. I had no way of checking this, and could not respond, because my own book - which as far as I knew, was unique of its kind - did not go beyond ten-figure primes. But Michael was up to it, though it took him five minutes - and an hour later the twins were swapping twenty-figure primes, at least I assume this was so, for I had no way of checking it. Nor was there any easy way, in 1966, unless one had the use of a sophisticated computer. And even then, it would have been difficult, for whether one uses Eratosthenes' sieve, or any other algorithm, there is no simple method of calculating primes. There is no simple method, for primes of this order - and yet the twins were doing it.”

Other quotes from the essay:

http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/twins.htm

 

[Mello_]

God is great mathmatician I must admit.

[+Desertrat56]

On 4/20/2020 at 6:43 PM, Mello_ said:

Sometimes after 6n you must add one or sub one.

Have you ever had a math class past the basics or are you trying to teach your self higher math?