Victor Reijkersz

Erm...since when is 17 composite?

 

Sorry for the delay in answering Vic, I completely forgot about this topic. I'm pretty tired at the moment, so I'm not all that excited about reading the whole thing again to remember what my argument was, but from what I do remember, your method requires remembering all the primes found thus far, to calculate the gaps (if not, I'm sorry, I really can't remember the method all that well). If that is the case, it is as effective, maybe even less, than the simplest algorithm for primes.

If on the other hand you wanted to prove that there was a connection between the distribution of primes and composites (I'm going to go on a whim and assume that's what you meant by "same way as"), and I'm not sure I see how your method proves that, that can be (I think) proven a lot faster. If you take the set of natural number and remove all primes, all that's left are composites (excluding the number 1). Furthermore, primes are the building blocks from which composite numbers are created, so I'd certainly expect there to be a connection between the two.

Please note however, I'm not an expert in this matter, far from it. Most of the above is just intuition working, so I apologize if it turns out to be gibberish. Also, I must say that's a pretty interesting way of generating primes. Thumbs up

 

thanks. to be honest i am pretty buisy at the moment too. so thank you too for taking the time so far.

 

i am limiting my discussion of this algorithm to

http://tech.groups.yahoo.com/group/primenumbers/messages

for the moment.

 

somebody there actually pointed out to me that my sieve method predicts everything correctly until the number 1331 that it will say its prime while its not.

i think i can repair it though if i had some good sleep and some more time

 

but i welcome you to join up on the prime mailing list. i think its a better place for discussion since its more focussed on the topic

 

best,

vic

 

 

 

Okay, that's what I though. In that case, my question is how do you calculate the individual slots?

 

the individual slots are equal to the gaps between the primes only for each prime we are calculating the slots for we have to shift 1 up.

 

 

so when calculating for the 1st prime (2) we start filling in the slots with first gap between primes (start at 1, the gap between 2 and 3).

gives the pattern [ 1 ]

 

when calculating for the 2nd prime (3) we start filling in the slots with second gap between primes (start at 2, the gap between 3 and 5).

gives the pattern [ 2 ]

 

when calculating for the 3rd prime (5) we start filling in the slots with third gap between primes (start at next 2, the gap between 5 and 7).

gives the pattern [ 2, 4 ]

 

when calculating for the 4th prime (7) we start filling out the slots with the fourth gap between primes (start at 4 , the gap between 7 and 11)

gives the pattern [ 4, 2, 4, 2, 4, 6, 2, 6 ]

I can see they are prime gaps, but how do you decide how many of them will play a role in the "pattern"? In the pattern for 7, you use 8 prime gaps, in the pattern for 11, you use 48 prime gaps..why?

Prime Number 2. pattern size 1

Prime Number 3. pattern size 1

Prime Number 5. pattern size 2

Prime Number 7. pattern size 8

Prime Number 11. pattern size 48

Prime Number 13. pattern size 480

the pattern size (or the number of slots) for a prime is equal to the pattern size of the previous prime multiplicated by the (previous prime minus 1).

for prime 3 its ((2-1)*1) = 1

for prime 5 its ((3-1)*1) = 2

for prime 7 its ((5-1)*2) = 8

for prime 11 its (7-1)*8) = 48

for prime 13 its (11-1)*48)= 480

etc..

 

And also, how do you determine the prime gaps in advance of knowing which numbers are prime?

 

Check the original post where i start at 2 and determine that 4 is a composite number because its the square of 2.

i then see that 3 is a number that has not been ruled a composite number by previous primes and i found the next prime.

Then arriving on 4, which we already know is a composite unique to 2, we know now the gap between the first 2 primes and can thus establish the pattern and pattern size for prime number 2 based on the info we now have.

the pattern size is of course 1 because its the first prime. and its one and only slot has value '1' because its the distance between the 1st and 2nd prime (2 and 3).

etc..

 

with the larger numbers we won't have found all the prime gaps neccessary to fill their slots, but thats not a problem because the patterns grow more rapidly then they are used up in having to determine new composite numbers.

 

again in the original post i take the reader by the hand in going through the sieve.

 

i hope this clarification again helps.

let me know if it does.

 

i tried this out on computer and it works. Up to a 6 digit numbers or so. This method finds the primes BEFORE it needs to know the gap sizes.

 

thanks again for taking the time to go over things. its really appreciated.

 

best regards,

Vic

 

 

I'm sorry, but I don't understand what "pattern size" and "slot values of patterns" are from your explanation.

 

 

Each prime has its own pattern.

For example for Prime7 its pattern is [ 4, 2, 4, 2, 4, 6, 2, 6 ]

 

Its pattern size is 8

The value of slot 1 of its pattern is 4

The value of slot 2 of its pattern is 2

etc...

the value of the last slot, slot 8, of its pattern is 6

 

The Pattern of Prime7 dictatates that the following numbers are composite numbers:

49,77,91,119,etc...

49 because its the square of 7

77 because its 49 + [ 4 * 7 ] (note that 4 is the 1st slot the pattern)

91 because its 77 + [ 2 * 7 ] (note that 2 is the 1st slot the pattern)

119 because its 91 + [ 4 * 7 ]

133 because its 119 + [ 2 * 7]

161 because its 133 + [ 4 * 7]

203 because its 161 + [ 6 * 7]

217 because its 203 + [ 2 * 7]

259 because its 217 + [ 6 * 7] note that 6 is the 8th slot the pattern)

...and the pattern repeats....

287 because its 259 + [ 4 * 7]

301 because its 287 + [ 2 * 7]

329 because its 329 + [ 4 * 7]

etc..

 

does this clear up what i mean with pattern, pattern size and slots of the pattern?

 

best regards,

Vic

I also posted the following in the sci.math newsgroup but got no responses that actually were interested in the main point of the "discovery" i thought i made.

would be happy if somewhere here could find the time to actually fill in the sieve of numbers as i am illustrating below:

Sieve of Vic?

I think i have discovered a more beautifull way to find the prime

numbers by using a Sieve. But i might be mistaken and have

rediscovered the wheel. My prime finding sieve method shows the

iterative nature of the primes very well and is therefore intriguing.

In short each primes causes an infinite number of other numbers to be

composite-numbers, but the composite numbers that are caused by each

prime are spread out in the exact same pattern as the primes

themselves are spread out.

 

 

I am not a mathimatican so please bare with me while I illustrate by

example instead of by formula. I would appreciate any serious

feedback. It might be I re-invented the wheel. I dont know. I dont do

maths often. only have been looking at primes as a sudoko puzzle. But

i thought i might actually have stumbled on an original thought. Hence

this post.

 

 

I am using a sieve approach for finding prime numbers. Just like

Eratosthenes. Noting all the numbers on a big sheet starting with 2

and numbering to however much you like.

 

 

Number 2 is the first prime in my mind. I note it on the primelist.

For every prime i find i have to cross out its power. 2 * 2 = 4. I

make a note on 4 that its the 1st composite-number caused by prime2.

 

 

Number 3 is not crossed out so its a prime. I note it on the

primelist. I now also cross out its power. 3*3=9. I make a note on 9

that its the 1st composite-number caused by prime3.

 

 

Number 4 is crossed out so its not a prime. Prime2 however left off

here. The pattern of prime 2 can now also be established. Its size is

1 because its the first pattern defined and its only pattern slot is 1

too. 1 because thats the difference between the first prime (2) and

the next (3). Knowing this pattern i know now the 2nd composite-number

caused by 2 must be current number(4) + (slot value(1) * prime(2)) =

6. I make a note on 6 that its the 2nd composite number caused by

prime2.

 

 

Number 5 is not crossed out so its a prime. I note it on the

primelist. I now also cross out its power 5*5 =25. I make a note on 25

that its the 1st composite-number caused by prime5.

 

 

Number 6 is crossed out so its not a prime. Prime2 however left off

here. Knowing this pattern i know now the 3rd composite-number caused

by 2 must be current number(6) + (slot value(1) * prime(2)) = 8. I

make a note on 8 that its the 3rd composite number caused by prime2.

 

 

Number 7 is not crossed out so its a prime. I note it on the

primelist. I now also cross out its power 7*7 =49. I make a note on 49

that its the 1st composite-number caused by prime7.

 

 

Number 8 is crossed out so its not a prime. Prime2 however left off

here. Knowing this pattern i know now the 4th composite-number caused

by 2 must be current number(8) + (slot value(1) * prime(2)) = 10. I

make a note on 10 that its the 4th composite number caused by prime2.

 

 

Number 9 is crossed out so its not a prime. Prime3 however left off

here. The pattern of prime 3 can now also be established. Its size is

1 because its the second pattern defined and its only pattern slot is

2. 2 because thats the difference between the second prime (3) and the

next (5).

Knowing this pattern i know now the 2nd composite-number caused by 3

must be current number(9) + (slot value(2) * prime(3)) = 15. I make a

note on 15 that its the 2nd compositie number caused by prime3.

 

 

Number 10 is crossed out so its not a prime. Prime2 however left off

here. Knowing this pattern i know now the 5th composite-number caused

by 2 must be current number(10) + (slot value(1) * prime(2)) = 12. I

make a note on 12 that its the 5th composite number caused by prime2.

 

 

Number 11 is not crossed out so its a prime. I note it on the

primelist. I now also cross out its power 11*11 =121. I make a note on

121 that its the 1st composite-number caused by prime11.

 

 

Number 12 is crossed out so its not a prime. Prime2 however left off

here. Knowing this pattern i know now the 5th composite-number caused

by 2 must be current number(12) + (slot value(1) * prime(2)) = 14. I

make a note on 14 that its the 6th composite number caused by prime2.

 

 

etc... etc... for number 13 and number 14

 

 

Number 15 is crossed out so its not a prime. Prime3 however left off

here. Knowing this pattern i know now the 3rd composite-number caused

by 3 must be current number(15) + (slot value(2) * prime(3)) = 21. I

make a note on 21 that its the 3rd compositie number caused by prime3.

 

 

..etc.. etc..

 

 

Number 25 is crossed out so its not a prime. Prime5 however left off

here. The pattern of prime 5 can now also be established. Its size is

2 because its the third pattern defined and its pattern size is 2 with

the pattern slots being [2, 4] because thats respectivly the

difference between the third prime (5) and the next (7) and the next

one(7) and the next-next one(11).

Knowing this pattern i know now the 3rd composite-number caused by 5

must be current number(25) + (slot value(2) * prime(5)) = 25. I make a

note on 35 that its the 2nd compositie number caused by prime5.

 

 

etc.. etc...

 

 

Number 35 is crossed out so its not a prime. Prime5 however left off

here. Knowing this pattern i know now the 4th composite-number caused

by 5 must be current number(35) + (slot value(4) * prime(5)) = 55. I

make a note on 55 that its the 4th compositie number caused by prime5.

 

 

etc..

 

 

Number 55 is crossed out so its not a prime. Prime5 however left off

here. Knowing this pattern i know now the 5th composite-number caused

by 5 must be current number(55) + (slot value(2) * prime(5)) = 65.

Since this the third slot value we have look up for 5 and the pattern

size was only 2 we return here to slot 1. I make a note on 65 that its

the 5th compositie number caused by prime5.

 

 

etc..

ad infinitum...

 

 

TABLE OF PATTERN SIZE

Prime2 = 1

Prime3 = 1 (1*1)

Prime5 = 2 ( 2*1)

Prime 7 = 8 (4*2)

Prime 11= 48 (6*8)

Prime 13 = 480 (10*48)

Prime 17= 5760 (12*480)

Prime 19 = 92160 (16*5760)

Next pattern size is thus based on the ((current prime - 1 ) *

current pattern size)

 

 

TABLE OF SLOT VALUES OF PATTERNS

Number 2. [ 1 ]

Number 3. [ 2 ]

Number 5. [ 2, 4 ]

Number 7. [ 4, 2, 4, 2, 4, 6, 2, 6 ] (8 numbers)

Number 11. [ 2, 4, 2, 4, 6, ... ] (48 numbers)

Number 13. [ 4, 2, 4, 6, .. ] (480 numbers)

 

 

TABLE OF COMPOSITE NUMBERS CREATED BY PRIMES

Prime2 : 4,6,8,10,12,14,etc..

Prime3 : 9,15,21,27,33,etc...

Prime5: 25,35,55,65,85,95,115,125,etc...

Prime7: 49,77,91,119,etc...

 

 

i can be contacted too at v...@xs4all.nl

 

 

thanks for any feedback,

Vic