Problems & Puzzles: Puzzles

Puzzle 812. Other chains of primes

Jeff Heleen sent the following nice puzzle:

Let P(n) and P(n+1) be consecutive primes. Let C(1)...C(m) be the composite numbers between P(n) and
P(n+1). If you sum the composites C(1)+...+C(m) and add 1 sometimes a prime, Q(k), results. Using primes Q(k) 
and Q(k+1) as new bounds, repeat the process. If another prime is found you have a second link in a chain.
 Continue iterating until a composite is reached. Twin primes sometimes lead to other primes and sometimes
 end there.

The smallest example is (5,7). Composite (6)+1=7. (7,11) yields (8+9+10)+1=28=composite.

 The smallest non-twin prime is 23 and 29. Composites (24+25+26+27+28)+1=131, a prime. The process ends 
 there because the next iteration (with bounding primes 131 and 137) produces 671, which is composite.

Here are some results:

(composite sum)+1
 1-link (twin): (5,7) -> 7 -> composite
1-link: (23,29) -> 131 -> composite
 2-link: (83,89) -> 431 -> 433 -> composite
 3-link: (53,59) -> 281 -> 283 -> 2593 -> composite
 4-link: (545477,545483) -> 2727401 -> 13637021 -> 68185121 -> 68185123 -> composite
 5-link: (16001,16007) -> 80021 -> 1360511 -> 6802571 -> 6802573 -> 61223203 -> composite

When subtracting 1 instead of adding, twin primes go into an

endless loop, so a twin prime is the end of this process.

Here are some results:

(composite sum)-1
 1-link: (73,79) -> 379 -> composite
 2-link: (421,431) -> 3833 -> 49919 -> twin prime
 3-link: (148873,148891) -> 2530993 -> 12654979 -> 265754789 -> twin prime

Q. Please obtain larger chains for each type of process.

Contributions came from Vicente F. Izquierdo and Emmanuel Vantieghem

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Vicente wrote:

Result for (composite sum)+1:

6-link: (1029201143,1029201149 )

Not result for 7-link for primes ≤ 2^32.

 

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Emmanuel wrote:

 

+1 case
 

6-link: (1029201143, 1029201149) -> 5146005731 -> 25730028671 -> 1054931176373 ->  5274655881881 -> 5274655881883 -> 47471902936993 -> composite

A 7-link should start above 2*10^9.


 

-1 case

4-link: (78712861, 78712871) -> 708415793 ->  34712375081 ->  867809377349 -> 11281521905627 -> composite

A 5-link should start above 2*10^9.

 

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