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Goldbach chains

Hi,

Here are some Goldbach chains:

Chains that are a multiple of 6 have more links so are longer, I don't think there can be an infinite length chain though.

The properties of a Goldbach chain:

ie for one of the 30 Goldbach chain with repeating elements removed:

7,30,53-53,60,67-67,120,173-173,240,307-307,480,653-end (5 links) (remove repeating links:) 7,30,53,60,67,120,173,240,307,480,653 (5 links)

then: for even i starting at zero: x[i]+x[i+2]=x[i+3] ie: 7+53=60 that is two primes that add to 2*30multiples

for odd i: (x[i]-x[i-1])+x[i-1]=x[i] for odd i: (x[i]-x[i-1]) is prime and (x[i]-x[i-1]) + x[i-1] are two primes that add to x[i] ie: (30-7)+7=23+7=30 that is two primes that add to 30 multiples

Here is a list of Goldbach chains up to n=35

n: Goldbach chain

0: none (total links=0)

1: none (total links=0)

2: none (total links=0)

3: none (total links=0)

4: none (total links=0)

5: one Goldbach chain (total links=2)

3,5,7-7,10,13-end (2 links)

6: none (total links=0)

7: none (total links=0)

8: two Goldbach chains (total links=3)

3,8,13-13,16,19-none (2 links) 5,8,11-end (1 link)

9: one Goldbach chain (total links=1)

7,9,11-end (1 link)

10: two Goldbach chains (total links=3)

3,10,17-17,20,23-end (2 links) 7,10,13-end (1 link)

11: none (total links=0)

12: two Goldbach chains (total links=7) 5,12,19-19,24,29-29,48,67-end (3 links) 5,12,19-19,24,29-29,96,163-end (2/3 links) --** note this is a skipped chain test 12,24,96 (48 link missing) -- didn't find any 163 to link up with at higher multiples.. 7,12,17-17,24,31-end (2 links)

13: none (total links=0)

14: one Goldbach chain (total links=1) 11,14,17-end (1 link)

15: one Goldbach chain (total links=2)

13,15,17-17,30,43-end (2 links)

16: two Goldbach chains (total links=2)

3,16,29-end (1 link) 13,16,19-end (1 link)

17: none (total links=0)

18: three Goldbach chains (total links=7) 5,18,31-31,36,41-41,72,103-end (3 links) 7,18,29-29,36,43-43,72,101-end (3 links) 13,18,23-end (1 link)

19: none (total links=0)

20: two Goldbach chains (total links=3) 3,20,37-37,40,43-end (2 links) 17,20,23-end (1 link)

21: one Goldbach chain (total links=1)

19,21,23-end (1 link)

22: one Goldbach chain (total links=2)

3,22,41-41,44,47-end (2 links)

23: none (total links=0)

24: five Goldbach chains (total links=9) 5,24,43-43,48,53-53,96,139-end (3 links) 7,24,41-end (1 link) 11,24,37-37,48,59-end (2 links) 17,24,31-end (1 link) 19,24,29-29,48,67-end (2 links)

25: none (total links=0)

26: one Goldbach chain (total links=1) 23,26,29-end (1 link)

27: none (total links=0)

28: none (total links=0)

29: none (total links=0)

30: five Goldbach chains (total links=16) 7,30,53-53,60,67-67,120,173-173,240,307-307,480,653-end (5 links) 13,30,47-47,60,73-73,120,167-167,240,313-313,480,647-end (5 links) 17,30,43-end (1 link) 19,30,41-41,60,79-end (2 links) 23,30,37-37,60,83-83,120,157-end (3 links)

31: none (total links=0)

32: one Goldbach chain (total links=2) 3,32,61-61,64,67-end (2 links)

33: none (total links=0)

34: one Goldbach chain (total links=1) 31,34,37-end (1 link)

35: none (total links=0)

36: five Goldbach chains (total links=) [not shown yet]

From looking at the 30 chain, I can see that the chains will always be finite I think, since even if there are lots of initial chains, they keep their relative spacing even while the primes are thinning out as the chains get longer, so the chain length has to be finite, but also the chains will be longest for large primorial multiples, ie 30 is the largest primorial in the above, and has 2 chains with 5 links each, and a total of 16 links counting all 5 chains, the most of all n up to 35 (all that were checked so far).

cheers, Jamie

Reply to
Jamie M
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A couple more properties of Goldbach chains:

ie for this example Goldbach chain:

7,30,53-53,60,67-67,120,173-173,240,307-307,480,653-end (5 links)

each of the sections is a link, and there are 5 links:

link1: 7,30,53 link2: 53,60,67 link3: 67,120,173 link4: 173,240,307 link5: 307,480,653

Each link has three elements call them a,b,c ie for link2:

a=53 b=60 c=67

For any link x, c-b = link x-1 a

ie the c-b value for link x=2 67-60=7 is equal to the previous links 'a' value, ie which for link x=1 is 7.

Also another property is that for each link with elements a,b,c (c-b) + a = b

ie for link2:

(67-60)+53 = 60

So these are hard to find long chains with these properties are all valid, but the longest chains are for where b is a large primorial number.

cheers, Jamie

Reply to
Jamie M

I decided to look for some longer Goldbach chains, so I started with a larger primorial multiple, to see if it really does have more Goldbach links. If not then it is possible the longest Goldbach chain is actually quite short.

So primorial 210 has 26 Goldbach chains, primorial multiple 420 has 38 Goldbach chains primorial multiple 840 has 59 Goldbach chains primorial multiple 1680 has 87 Goldbach chains primorial multiple 3360 has 140 Goldbach chains primorial multiple 6720 has 200 Goldbach chains etc..

So what is the longest chain starting from 210?

Here is the list of 26 chains and their number of links.

1: 11,210,409-409,420,431-431,840,1249-1249,1680,2111-end (4 links)

2: 19,210,401-401,420,439-end (2 links)

3: 31,210,389-end (1 link)

4:

37,210,383-383,420,457-457,840,1223-1223,1680,2137-2137,3360,4583-end (5 links)

5: 47,210,373-373,420,467-467,840,1213-end (3 links)

6: 53,210,367-end (1 link)

7: 61,210,359-end (1 link)

8: 71,210,349-349,420,491-end (2 links)

9: 73,210,347-end (1 link)

10: 83,210,337-337,420,503-end (2 links)

11: 103,210,317-317,420,523-end (2 links)

12: 107,210,313-end (1 link)

13: 109,210,311-end (1 link)

14: 113,210,307-end (1 link)

15: 127,210,293-293,420,547-end (2 links)

16: 137,210,283-283,420,557-557,840,1123-1123,1680,2237-2237,3360,4483-end (5 links)

17: 139,210,281-end (1 link)

18: 149,210,271-271,420,569-end (2 links)

19: 151,210,269-269,420,571-571,840,1109-1109,1680,2251-end (4 links)

20: 157,210,263-263,420,577-577,840,1103-end (3 links)

21: 163,210,257-end (1 link)

22: 179,210,241-241,420,599-end (2 links)

23: 181,210,239-239,420,601-end (2 links)

24: 191,210,229-end (1 link)

25: 193,210,227-227,420,613-end (2 links)

26: 197,210,223-223,420,617-617,840,1063-1063,1680,2297-2297,3360,4423-end (5 links)

So the longest Goldbach chain starting from 210 has 5 links, the same length as the longest Goldbach chain starting from 30.

Starting from 210 there are 3 Goldbach chains that have 5 links, and starting from 30 there are 2 Goldbach chains that have 5 links:

from 210:

37,210,383-383,420,457-457,840,1223-1223,1680,2137-2137,3360,4583

137,210,283-283,420,557-557,840,1123-1123,1680,2237-2237,3360,4483

197,210,223-223,420,617-617,840,1063-1063,1680,2297-2297,3360,4423

from 30:

7,30,53-53,60,67-67,120,173-173,240,307-307,480,653

13,30,47-47,60,73-73,120,167-167,240,313-313,480,647

I am not sure if there are any longer chains than length=5

cheers, Jamie

Reply to
Jamie M

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