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OT: can someone explain this pattern in the primes

Hi,

Here is a new pattern I found in the number arrays that is quite cool I think!

ie number array 5:

0 10 1 9 2 8 3 7 4 6 5 5

That number array has two prime pairs 3,7 and 5,5

The offset from the last row of the number array to these prime pairs is 2 for 3,7 and 0 for 5,5 which makes a sequence 2,0

This sequence (call it the "prime pair gap sequence") has one prime, 2.

Ok so that seems meaningless BUT.. here is the new pattern..

for number arrays that are multiples of 30, ie number arrays

30,60,90,120... guess what?!

These number arrays have the most primes in their "prime pair gap sequences" of any other number arrays by a large margin, and as the number array size increases by a multiple of 30, the margin of increased primes seems to go up too compared to other number arrays of similar size.

ie

number array 30:

0 60 1 59 2 58 3 57 4 56 5 55 6 54 7 53 primepair 8 52 9 51 10 50 11 49 12 48 13 47 primepair 14 46 15 45 16 44 17 43 primepair 18 42 19 41 primepair 20 40 21 39 22 38 23 37 primepair 24 36 25 35 26 34 27 33 28 32 29 31 primepair 30 30

Ok so you can see that based on the left column, the primepairs seem to occur at 7,13,17,19,23,29 no big deal right?

BUT GUESS WHAT!?

If you could from the bottom like this:

30 0 60 29 1 59 28 2 58 27 3 57 26 4 56 25 5 55 24 6 54 23 7 53 primepair 22 8 52 21 9 51 20 10 50 19 11 49 18 12 48 17 13 47 primepair 16 14 46 15 15 45 14 16 44 13 17 43 primepair 12 18 42 11 19 41 primepair 10 20 40 9 21 39 8 22 38 7 23 37 primepair 6 24 36 5 25 35 4 26 34 3 27 33 2 28 32 1 29 31 primepair 0 30 30

Now the prime pairs, when counted using the leftmost reversed column, occur at:

1,7,11,13,17,23

and 5 of 6 of these are primes.

This pattern of reversible prime occurrence in the primepairs occurs for multiples of 30 number arrays, so the next big pattern is for number array 60.

Here are the reversed prime pair patterns for some of the multiples of 30:

number array 30 (5 primes out of 6 numbers):

7,13,17,19,23,29

number array 60 (10 primes out of 12 numbers):

7,11,13,17,19,23,31,37,41,47,53,59

number array 90 (12 primes out of 14 numbers):

7,13,17,23,29,31,41,43,53,67,71,73,79,83

number array 120 (16 primes out of 18 numbers):

7,11,13,17,29,41,43,47,59,61,67,73,83,89,101,103,109,113

number array 210 (26 primes out of 30 numbers):

11,19,23,31,37,41,47,53,61,67,71,73,83,89,103,107,109,113,127,137,139,149,151,157,163,179,181,191,193,197

number array 420 (38 primes out of 51 numbers):

11,13,17,19,29,31,43,53,67,71,79,83,89,97,101,107,113,131,139,149,157,163,167,179,181,193,197,199,223,227, 233,239,241,263,269,271,277,283,293,317,331,337,349,353,373,379,383,397,401,409,419

Here is a spreadsheet showing this stuff for number arrays up to 499, I put the "60 example" in there on the right of the spreadsheet highlighted in yellow.

There might be patterns like this for primorial multiples like multiples of 210 etc I think, but haven't checked yet.

Link to the spreadsheet:

formatting link

cheers, Jamie

Reply to
Jamie M
Loading thread data ...

On Mon, 4 Apr 2016 22:31:54 -0700, Jamie M Gave us:

WRONG GROUP, STUPID FUCK!

Reply to
DecadentLinuxUserNumeroUno

On 4/4/2016 10:37 PM, Jamie M wrote: > Hi, > > Here is a new pattern I found in the number arrays that is quite > cool I think! >

Ya it looks like it is a pattern of +- primes and centered on the primorials, out of the first 2651 number arrays, primorial 2310 has the highest count of prime pairs with prime number offsets

+- from 2310 out of all the numbers.

Partial list showing the arrays from 0 to 2651 with the most prime pair gaps that are prime numbers:

numberArray PrimePairCountGapsThatArePrimeNumbers

2310 124 2520 99 2100 98 2640 95 1890 89 1680 87 2340 87 2280 82 2580 81 1470 79 1980 79 2610 78 2550 77 2040 76 2370 76 2430 75 1260 74 2070 74 2400 73 1650 72 2460 72 2490 72 1950 71 2160 69 2220 69 2130 68 1860 67 2010 66 1560 65 1710 65 1740 65

I guess this +- of a primorial is a thing in sieve theory?

cheers, Jamie

Reply to
Jamie M

On Tue, 5 Apr 2016 00:38:07 -0700, Jamie M Gave us:

Go the f*ck away! You pathetic, group interloping dumbfuck.

Reply to
DecadentLinuxUserNumeroUno

Be gentle. He may have a mental problem.

Reply to
John S

At least *one* of them certainly has.

Reply to
Julian Barnes

At least two >:-} ...Jim Thompson

--
| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
| Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    | 
| San Tan Valley, AZ 85142   Skype: Contacts Only  |             | 
| Voice:(480)460-2350  Fax: Available upon request |  Brass Rat  | 
| E-mail Icon at http://www.analog-innovations.com |    1962     | 
              
           The touchstone of liberalism is intolerance
Reply to
Jim Thompson

Is there something useful here ??

...Jim Thompson

--
| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
| Analog/Mixed-Signal ASIC's and Discrete Systems  |    manus    | 
| San Tan Valley, AZ 85142   Skype: Contacts Only  |             | 
| Voice:(480)460-2350  Fax: Available upon request |  Brass Rat  | 
| E-mail Icon at http://www.analog-innovations.com |    1962     | 
              
           The touchstone of liberalism is intolerance
Reply to
Jim Thompson

What does it look like in octal, or Hex?

RL

Reply to
legg

Hi,

I think it doesn't matter what base you use for this analysis, since it is a pattern that exists around the primorials that are inherent to the primes, so there should be a similar pattern in any base I would think.

Good to check though.

cheers, Jamie

Reply to
Jamie M

Here is a picture of a graph showing the data for all numbers up to about 2500. The Y axis is the number of prime pairs +- for each number that have a prime gap from the number, and x axis are the numbers from

0 to 2651 that have at least 3 prime pairs +- of the number that occur with prime offsets.

Primorial multiples make up the scattered dots with highest y values in the graph as those numbers have the most prime pairs +- of the number that occur with prime offsets.

formatting link

Also here is a graph showing all numbers 0 to 2651, even those that have less than 3 prime pair multiples etc.. quite a few numbers have no prime pairs +- that occur with prime offsets which are the horizontal line at y=0.

formatting link

With increasing the number range up past 30030 etc to get more primorial 2310 multiples, there should be some more groupings of scatter points visible in the graphs, but right now it shows the pattern too.

cheers, Jamie

Reply to
Jamie M

Yessir! But whether it is anything new or not is another question, but it is definitely a pattern in the primes that is really neat.

I can try to explain it using the numbers 0 to 60:

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29, 30, 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60

The numbers are written in three rows like that to show that the number

30 is the center point of the prime pattern (and all multiples of 30 and higher primorials are the center points of similar patterns)

So for the 30 pattern, and other center point numbers, on either side of

30, ie 0 to 29 for one side, and 31 to 60 as the other side:

(0 to 29) 30 (31 to 60)

the 0 to 29 side has a similar pattern of prime distribution, relative to the center point 30, as the 31 to 60 side does, and this pattern holds for all of these center point numbers (primorials and all primorial multiples, ie 30, 60, 90,..., 210, 420, etc..)

So for 30 center point for this set of primes on the left side of 30, in the range 0 to 29:

7,13,17,19,23,29

There are corresponding primes on the right side of 30, in the range

31 to 60, they have a formula using the center point 30 that relates them to the left side primes: (30-leftSidePrime)+30

(30-7)+30 (30-13)+30 (30-17)+30 (30-19)+30 (30-23)+30 (30-29)+30

That gives the right side primes:

53,47,43,41,37,31

reversed that is just:

31,37,41,43,47,53

So now there is this pattern:

7,13,17,19,23,29 [30 center point] 31,37,41,43,47,53

So these primes gaps from 30 are equal on both sides:

29 and 31 are both 1 number away from 30 23 and 37 are both 7 numbers away from 30 19 and 41 are both 11 numbers away from 30 17 and 43 are both 13 numbers away from 30 13 and 47 are both 17 numbers away from 30 7 and 53 are both 23 numbers away from 30

So those common gaps away from 30 make the sequence:

1,7,11,13,17,23 which is all prime numbers.

I tested different center point numbers, and numbers like 30, 60, 90, 210, 2310, etc (the primorials) have the most prime numbers like this of any other numbers.

It is a fractal pattern of prime distribution with primorial number centerpoints I think.

cheers, Jamie

Reply to
Jamie M

The pattern I am asking about is how the multiples of 30 have gaps that are prime numbers. Other numbers that aren't multiples of 30 also have the prime pairs but they have far fewer occurrences of the prime pairs having prime number gaps.

Here is some data that should make it more clear:

number 29 has four prime pairs on either side of it within the range 0 to 58 (including 29 as one of the prime pairs)

5,11,17,29 (29) 29, 41, 47, 53

These are called "prime pairs" since they have matching gaps to the center point 29. ie

29 leftside and 29 rightside both have gap=0 to center point 29 17 leftside and 41 rightside both have gap=12 to center point 29 11 leftside and 47 rightside both have gap=18 to center point 29 5 leftside and 53 rightside both have gap=24 to center point 29

**This makes a sequence of prime pair gaps: 24,18,12,0**

This sequence is where the pattern that I am asking about is. Since for the other example of centerpoint 30, and multiples of 30 (and primorials and multiples of primorials in general) these sequences of prime pair gaps tend to be prime numbers, far more often than other numbers ie 29.

Here is a bit of info you can see the multiples of 30 (and 6 since it is a primorial) have these sequences of primes, and you can see the sequence for 29 in there too for reference.

number primePairGapSequence

3 0 4 1 5 2,0 6 1 7 4,0 8 5,3 9 4,2 10 7,3 11 8,6,0 12 7,5,1 13 10,6,0 14 9,3 15 8,4,2 16 13,3 17 14,12,6,0 18 13,11,5,1 19 12,0 20 17,9,3 21 16,10,8,2 22 19,15,9 23 20,18,6,0 24 19,17,13,7,5 25 22,18,12,6 26 21,15,3 27 20,16,14,10,4 28 25,15,9 29 24,18,12,0 ***here is the number 29 used in above example*** 30 23,17,13,11,7,1 31 28,12,0 32 29,27,21,15,9 33 28,26,20,14,10,4 34 27,3 35 32,24,18,12,6 36 31,25,23,17,7,5 37 34,30,24,6,0 38 35,33,21,15,9 39 34,32,28,22,20,8,2 40 33,27,21,3 41 38,30,18,12,0 42 37,31,29,25,19,11,5,1 43 40,36,30,24,0 44 39,27,15,3 45 38,34,28,26,22,16,14,8,2 46 43,33,27,15 47 42,36,24,6,0 48 41,35,31,25,19,11,5 49 30,18,12 50 47,39,33,21,9,3 51 46,38,32,28,22,20,10,8 52 49,45,21,15,9 53 50,48,36,30,6,0 54 49,47,43,35,25,17,13,7 55 52,48,42,24,18,12 56 53,51,45,33,27,15,3 57 52,50,46,44,40,26,16,14,10,4 58 55,51,45,39,21,15 59 54,48,42,30,12,0 60 53,49,47,43,41,37,29,23,19,13,7,1 61 48,42,18,0 62 51,45,39,21,9

See multiples of 6 and 30 have primes.

This is a fractal pattern with primorial center points as the "strongest" pattern, and then diminishing pattern of these prime sequences for numbers that are less like primorials. Multiples of primorials are similar to primorials so they have the pattern.

cheers, Jamie

Reply to
Jamie M

Here you can see the periodic fractal pattern that will only increase as more numbers are added. There are 6 subpeaks for every cycle of the largest peaks, and once multiples of larger primorials show up with bigger numbers, the biggest peaks in this chart will become subpeaks too, so it is a fractal pattern.

The peaks are primorials and multiples of primorials, the x axis is number 1 to 7142, and the y axis is the number of prime pairs for each number that are prime.

formatting link

full spreasheet if you want to scroll through all the numbers to see the full pattern:

formatting link

cheers, Jamie

Reply to
Jamie M

Restated mathematically:

Prime pair of n definition: For a given number n and for two primes x and y, with x between n-n and n and y between n and n+n, if n-x is equal to y-n then x and y are called a prime pair z(x y) of n.

Primal pair: For prime pair z(x y) if n-x is a prime number then prime pair z is called a primal pair Z(x y). It follows if n-x is a prime number then y-n is also a prime number as y-n = n-x

Primorial numbers n, ie 6, 30, 210, 2310, 30030 and to a lesser extent multiples of primorials ie 30,60,90,..240, etc have more primal pair's Z associated with them than any other numbers.

The count of primal pair's Z increases with larger primorial numbers n.

The distribution of the primal pair's Z for a given n occurs in the range n-n to n+n, in other words every number n can have primal pairs in the range from zero to n+n, which is why for larger numbers the count of primal pair's Z increases.

Reply to
Jamie M

The chart shows what I was talking about before, about primes being organized into hierarchical blocks:

formatting link

The chart has peaks between each block of primes, and the peaks are higher when between bigger blocks, blocks are primorial sized 6,30,210 etc, the block edges are at multiples of the primorials and that is where the peaks in the graph occur.

I wrote this before in the thread:

"simple proof to explain the shocking news about primes last digits not being random"

Basically the primes are organized into blocks!!

The block sizes are all length of primorials and they are all lined up with common edges on the blocks..

so for example there are 5 primorial6 blocks within primorial30:

((6)(6)(6)(6)(6)) = (30) a primorial30 block

and there are seven primorial 30 blocks in the next up primorial210

((30)(30)(30)(30)(30)(30)(30)) = (210) a primorial210 block

this goes on for infinity for all primorials.

Why is this important?

Because reason1 related to the recent news about primes last digits not being random:

The probability of the primes last digits is proportional to the block size considered! This is because each block size has a corresponding multiple of 4 set of equations that produce all the primes, except for the blocks (primorial) unique prime factors, and each of the equations will produce primes ending in a specific digit 1,3,7,9. So everytime a 1,3,7 or 9 ending prime is in a given block, the chance that there will be another 1,3,7, or 9 in that same block will be decreased proportional to the number of equations in the block that produce primes with that last digit

1,3,7 or 9.

As I stated before ie for primorial 2310 has 120 equations that produce primes ending in each of 1,3,7,9 for a total of 480 equations. So the probability is 120/480 = 25% for each prime in the range 2310 to be either 1,3,7,9. If there is a second prime in that 2310 range, then there is only a 24.79% chance (119/480) that it will also end with the same digit, since there are only 120 equations that can make primes for each ending digit in the 2310 number range.

Remember all the blocks are fractally consistent too! So it should be easily analyzed to find the probabilities of consecutive primes ending digits or any probability of repeating primes end digits etc.

Reply to
Jamie M

Okay, show us the numbers in base Pi.

Reply to
John S

Apple or Butterscotch? ;-)

Reply to
Michael A. Terrell

Of course!

Reply to
John S

Huge breakthrough 181:

As usual this breakthrough is related to the prime numbers.

And more specifically, it allows for a method to find likely primes from n to 2n, by doing a little searching from 0 to n.

Also the neat thing is that if you search 0 to n close to zero, then that corresponds to the primes that can be found close to 2n..

So say n is a giant primorial with 1000 digits. Then if I search the first 1% of those digits, and find x z-type prime pair candidates, I now know where to check for primes in the last 1% of 2n.

There is a statistical likelihood that each x is a Z-type prime pair, and if so it will have a matching prime number near 2n.

So what is the z-type and Z-type prime pairs?

Here are the definitions:

z(x y) Prime pair of n definition: For a given number n and for two primes x and y, with x between 0, and y between n and 2n, if n-x is equal to y-n, then x and y are called a prime pair z(x y) of n.

Z(x y) Primal pair: For prime pair z(x y) if n-x is a prime number then prime pair z is called a primal pair Z(x y). It follows if n-x is a prime number then y-n is also a prime number as y-n = n-x

For n = 30030, there are 852 Z-type prime pairs, so that means there are 852 primes that can be found between n and 2n, by choosing the correct z-type prime between 0 and n then checking the corresponding locations between 2n and n.

So how many z-type candidates are between 0 and n? I didn't check that yet so can't say what type of probability this gives for finding primes. But it looks like an interesting way to find big primes.

cheers, Jamie

Reply to
Jamie M

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