Problems & Puzzles: Puzzles

 Puzzle 1084 Chains related to twin prmes Paolo Lava sent the following nice puzzle: A) Consider twin primes p and (p+2) such that  p + (p+1) and (p+1) + (p+2) are twin primes too. Here below the first ten:   (5, 7), (11, 13) (29, 31), (59, 61) (659, 661), (1319, 1321) (809, 811 ), (1619, 1621) (2129, 2131), (4259, 4261) (2549, 2551), (5099, 5101) (3329, 3331), (6659, 6661) (3389, 3391), (6779, 6781) (5849, 5851), (11699, 11701) (6269, 6271), (12539, 12541).   Now, which are the least chains of n twin primes under this rule?   1) (3, 5) 2) (5, 7), (11, 13) 3) (211049, 211051), (422099, 422101), (844199, 844201) 4) (253679, 253681), (507359, 507361), (1014719, 1014721), (2029439, 2029441)   Q. Send other least chains for n>4 ?

During the week 23-29 May, 2022,contributions came from Giorgos Kalogeropoulos, Oscar Volpatti, Emmanuel Vantieghem.

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Giorgos wrote:

5) {41887255409,41887255411},{83774510819,83774510821},{167549021639,167549021641},{335098043279,335098043281},{670196086559,670196086561}

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Oscar wrote:

I found the least chains with length 5 and 6:
n = 5, p = 41887255409;
n = 6, p = 73768891456259;
where (p,p+2) is the starting twin pair.

Let's define parameter x = p+1; a chain of length n has pairs of the form (x*2^k-1,x*2^k+1), with 0<=k<=n-1. Looking at divisibility constraints modulo small primes:
11# divides x for n >= 5,
13# divides x for n >= 6.
Also dropping bad residues modulo primes up to 100, I sieved all p below 10^17, finding no chain with length 7 or more.

However, these "bi-twin" chains have been searched extensively. Thanks to Paul Jobling, Phil Carmody, Jaroslaw Wroblewski, solutions with length up to 9 are known:

n = 7, p = 337190719854678689,
n = 8, p = 631627448582856455729,
n = 9, p = 112511682470782472978099.

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Emmanuel wrote:

The least chain of five members is:
{41887255409,41887255411},{83774510819,83774510821},{167549021639,167549021641},{335098043279,335098043281},{670196086559,670196086561}
And this is the least chain of six members :
{73768891456259,73768891456261},{147537782912519,147537782912521},{295075565825039,295075565825041},{590151131650079,590151131650081},{1180302263300159,1180302263300161},{2360604526600319,2360604526600321}

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