Problems & Puzzles: Puzzles

Puzzle 738. Special Cunningham Chains Abhiram R Devesh sent the following puzzle: I had proposed 2 special Cunningham chains in OEIS a. A236443 - Primes which start a Cunningham chain of length 4 where every entity of the chain is smallest of its twin prime.    b. A237017 - Primes which start a Cunningham chain of length 4 where every entity of the chain is smallest of the prime number pair (p, p+8). So far i have searched till 8.5 E+10 (for 3days) and could not find any number common between these 2 series Q. can you find some common terms, or show that they do not exist?

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Contributions came from Jan van Delden, J. K. Andersen and Emmanuel Vantieghem

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Jan wrote:

There are no common terms.

The first type [p,p+2,2p+1,2p+3,4p+3,4p+5,8p+7,8p+9] implies p=209 mod 210.
The second type [p,p+8,2p+1,2p+9,4p+3,4p+11,8p+7,8p+15] implies p=89,149 or 179 mod 210.

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Andersen wrote:

A236443 is called a BiTwin chain of length 4, or with 3 links at
http://www.primenumbers.net/Henri/us/NouvChPus.htm
The largest known is at http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm:
223818083*409#*2^n +/- 1 for n=6,7,8,9, found by Dirk Augustin in 2006.
It has 177 digits. Torbjörn Alm and I had the previous 142-digit record.

OEIS already had the sequence as A069176(n) = A236443(n) + 1.

There are no common terms with A237017. The combined pattern
is inadmissible modulo 7. If the BiTwin chain has length at
least 3 then each p+1 and therefore p+8 is divisible by 7.

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Emmanuel wrote:

There is no common term for the sequences  A236443 (S1)  and  A237017 (S2).

Proof :  p  belongs to  S1  only if  p, p+2, 2p+1, 2p+3, 4p+3, 4p+5, 8p+7 and 8p+9  all are prime.

Looking at these numbers modulo 7, the only possible choice for  p  must be p = 6 mod 7.
 

For  p  to belong to  S2, p, p+8, 2p+1, 2p+9, 4p+3, 4p+9, 8p+7 and  8p+15 must be prime. Looking at these numbers modulo 7, the only possible choices for  p  must be  p = 2, 4, 5 mod 7. Hence, the intersection of  S1  and  S2  is empty, QED.

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