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

Here is some more info on this:

For equation sets for primorial 210, there are 48 equations that produce all the primes (except the primes 2,3,5,7 which are the unique prime factors of primorial 210). So that is 48/4 = 12 equations each that produce all primes ending in 1,3,7,9)

It is interesting to see the pattern in consecutive primorials: primorial 6 has two equations, 6x+1 and 6x+5, primorial 30 has 8 equations, primorial 210 has 48 equations, primorial 2310 has 480 equations, primorial 30030 has 5760 equations, and notice that

(primorial x equation count) / (primorial x-1 equation count) = pattern

48/8=6 480/48=10 5760/480=12 92160/5760=16 *extrapolated from guessed oeis sequence A005867 1658880/92160=18 36495360/1658880=22 1021870080/36495360=28

So there should be a pattern of how many equations there will be for any primorial, and the number of prime producing equations is always divisible by 4 to equally produce primes ending in 1,3,7,9, except for the primorial's unique prime factors.

So it looks like this sequence:

is how many equations each primorial has to produce all primes, and then this sequence: (.. 6, 10, 12, 16, 18, 22, 28,..) is the (primorial x equation count) / (primorial x-1 equation count), and this sequence happens to be Primes minus 1.

sequence

1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000

sequence

(primes minus 1)

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70,..

There are no equations in the same format that have fewer equations than the primorials for other numbers small than the primorial used, and as the primorial size gets larger the percentage of required equations to produce all the primes decreases and starts to just look like the sequence of primes.

cheers, Jamie

If primes were randomly distributed.

So you proving they're not randomly distributed by assuming that they are.

Sylvia.

The reason there is OEIS sequence: A006093 (primes minus 1)

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70,..

is because for the next larger primorial sequence, there is one more prime factor that isn?t included in the list of equations I think.

cheers, Jamie

Hi,

Proving that the last digits of the primes aren't randomly distributed.

cheers, Jamie

>

To evaluate the number of pairs of consecutive primes you could take a sum of all primorial equation sets equation counts, ie how it can work for just primorial 30:

y = 30x + 1 (produces half of primes ending in 1) y = 30x + 7 (produces half of primes ending in 7) y = 30x + 11 (produces half of primes ending in 1) y = 30x + 13 (produces half of primes ending in 3) y = 30x + 17 (produces half of primes ending in 7) y = 30x + 19 (produces half of primes ending in 9) y = 30x + 23 (produces half of primes ending in 3) y = 30x + 29 (produces half of primes ending in 9)

Within any 30 numbers, there is a 25% chance (2/8) for each prime to be a 1,3,7 or 9. But if there is already a 1,3,7, or 9 in that set of 30 numbers, and if there is a second prime in that 30 number range, then there is only a 14.28% chance (1/7) that it will also end with the same digit, since there are only 2 equations that can make primes for each ending digit in the 30 number range.

This has to be summed for all primorial equation sets to get the probability of finding the next prime for what digit it will end with. Each primorial equation set deals with a range of numbers that are equal to the primorial number itself. For large primorials there are many different equations that produce primes ending in 1,3,7,9, ie 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.

What I've been trying to figure out for awhile is how many equations a given primorial uses to create all the primes (except for the unique prime factors of the primorial used)

for primorial 6 it is 2 equations:

6x+1 6x+5

for primorial 30 it is 8 equations:

y = 30x + 1 y = 30x + 7 y = 30x + 11 y = 30x + 13 y = 30x + 17 y = 30x + 19 y = 30x + 23 y = 30x + 29

for primorial 210 it is 48 equations for primorial 2310 it is 480 equations for primorial 30030 it is 5760 equations

I found the pattern by checking on OEIS

This sequence is how many equations each primorial has to produce all primes:

sequence

1, 1, 2, 8, 48, 480, 5760, 92160, 1658880, 36495360, 1021870080, 30656102400, 1103619686400, 44144787456000, 1854081073152000, 85287729364992000, 4434961926979584000, 257227791764815872000, 15433667505888952320000

This sequence is (primorial x equation count) / (primorial x-1 equation count):

sequence

(primes minus 1)

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70,..

So matching up equation A005867:

primorial (primorial equation count equal to A005867 sequence)

6 2 30 8 210 48 2310 480 30030 5760 510510 92160 9699690 1658880 223092870 36495360 6469693230 1021870080 200560490130 30656102400 7420738134810 1103619686400 304250263527210 44144787456000 13082761331670030 1854081073152000

The reason I wanted to find this is to see the ratio of equations required as the size of primorial goes up, so now I have the equation it is easy, before by running my computer it took days to manually calculate the ratios just up to 2000 or so..

ie: these below ratios are related to the prime density of the primorial equations, the lower the ratio the higher the prime density.

the ratio of equations for primorial 6 is 2/6=0.3333.. the ratio of equations for primorial 30 is 8/30=0.2666.. the ratio of equations for primorial 210 is 48/210=0.2285.. the ratio of equations for primorial 2310 is 0.20779.. the ratio of equations for primorial 30030 is 0.1918.. the ratio of equations for primorial 510510 is 0.1805.. the ratio of equations for primorial 9699690 is 0.1710.. the ratio of equations for primorial 223092870 is 0.1636.. the ratio of equations for primorial 6469693230 is 0.1579.. the ratio of equations for primorial 200560490130 is 0.1529.. the ratio of equations for primorial 7420738134810 is 0.1487.. the ratio of equations for primorial 304250263527210 is 0.1451.. the ratio of equations for primorial 13082761331670030 is 0.1417..

Seems like the ratio is getting smaller and smaller, and it will approach zero eventually I think at infinite primorial size.

So what is the formula for the curve this sequence makes?

0.3333 0.2666 0.2285 0.20779 0.1918 0.1805 0.171 0.1636 0.1579 0.1529 0.1487 0.1451 0.1417 ... to zero I think

cheers, Jamie

Am 16.03.2016 um 14:53 schrieb Jamie M:

Did you notice that eulerphi(3) %561 = 2

eulerphi(3*5) %562 = 8

eulerphi(3*5*7) %563 = 48

eulerphi(3*5*7*11) %564 = 480

Gottfried

Hi,

No what does that signify?

I made another discovery though! :D

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.

cheers, Jamie

The key point is that it isn?t about consecutive primes, it is about the likelihood of subsequent primes with the same last digits occurring within a given primorial block 6,30,210,2310 etc, so it is a more generalized idea than just consecutive prime last digit occurrences.

What is the %561 is that mod?

cheers, Jamie

I see ya the number of equations that produce primes (other than the primorials unique prime factors) for a given primorial is eulerphi(product of the primorial primefactors except 2) I guess?

cheers, Jamie

Maybe that is all the coprimes of the primorial/2 !? :)

's_totient_function

cheers, Jamie

So maybe the formula is:

eulerphi(primorial/2) = number of equations that produce primes (besides the primorials unique prime factors)

so

eulerphi(6/2)=2

2 formulas 6n+1 and 6n+5

seems to work fine.

So with this easy formula its pretty easy to give the probability of primes occurring based on their last digits in any given primordial block range.

cheers, Jamie

Hi,

The chance of two primes with last digit 7 occuring in primorial range

30, the odds of that happening are (2/8 + 1/7)/2= 0.196

Proof:

The probability of the first prime being a 7 is 2/8 for primorial30 equation set, with eulerphi(30/2)=8 equations, two of them having numbers with least significant digits of 7. The odds of the second prime being also having a least significant digit of 7 is 1/7 as there are only 7 equations left that can produce primes in the primorial30 range, and only one equation that produces primes with significant digits of 7.

So the overall odds of having two primes with last digit being

7 within a primorial30 range is [(2/8)+(1/7)]/2=0.196428..

NOTE: that the Maximum number of primes that can have last digit of 7 (or more generally the same last digit) within a range of 30 numbers that are counted consecutively from zero, is 2.

The same calculation can be done for larger primorials, which can have more of the same ending digit primes in their primorial ranges counted from zero, but at decreasing probability for each subsequent one, using that above formula.

I still have to verify this 0.196428 is accurate :D

cheers, Jamie

And you felt it necessary to use the ?S? word, just like every spam pimp does?

How many equations are you saying are required for y = 210x + b to produce all the primes for non negative integers x?

Not counting 2,3,5,7 unique prime factors of the primorial 210, there are 48 different values of b required.

Do you know how to find all the values of b?

cheers, Jamie

Because 2*3*5 = 30, these equations produce all naturals greater than

30 that do not have 2 or 3 or 5 as a factor, including all such primes and all such nonprimes.

There is nothing magical or mystical or impressive about it!

The set of equations y = 6x + a prime less than 6 will produce every prime greater than 6

The set of equations y = 210x + a prime less than 8 will produce every prime greater than 8

The set of equations y = 2310x + a prime less than 12 will produce every prime greater than 12

And so on

--
Virgil 
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

Ya I am spamming I admit, but at least it is confined to one thread mainly.

I put shocking in quotation marks as I don't think it is shocking..

"Mathematicians shocked to find pattern in ?random? prime numbers"

note the lack of quotation marks from the media.

Gotta compare this result with the 17.757% result quoted from this page about this prime spacing:

"If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7? ...they found that among the first hundred million primes, the answer is just 17.757%."

I think the difference from 19.6% to 17.757% might be explained by the impact of subsequent periods of primorial ranges, not sure though.