Problems & Puzzles: Puzzles

 Puzzle 801. A nice stepladder of primes Vic Bold propose to find large steps of primes using the following scheme: Scheme #1 p+np(p)+1=q q+np(q)+1=r and so on... where np(x) is the next larger prime to x. I add the following scheme: Scheme #2 p+np(p)-1=q q+np(q)-1=r and so on... Q. Find the prime p generating the largest stepladder you may produce using both schemes.

Contributions came from Abhiram R. Devesh, Jeff Heleen, Dmitry Katmenesky and Emmanuel Vantieghem

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Abhiram wrote:

The largest prime I can find that can produce 4 step primes with both the schemes is 818625751

p=q+np(q)-1
818625751, 1637251541, 3274503083, 6549006173, 13098012353
p=q+np(q)+1
818625751, 1637251543 , 3274503091, 6549006199, 13098012451

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Jeff wrote:

For puzzle 801 I found these for the earliest instance of n steps where pi is not already part of an earlier sequence.

 +1 steps, n pi 1 2 3 4 5 6 0 3 1 29 61 2 5 13 31 3 23 53 113 241 4 17 37 79 163 331 5 111949 223903 447823 895651 1791319 3582643 6 4035869 8071757 16143541 32287093 64574197 129148399 258296803

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Dmitry wrote:

For both schemes I was able to find primes that produce stepladders of 9 primes (including the first one).

+1 version: 48558202601, 65310442037, 153993046523

-1 version: 20178627149, 41425492699, 42539054641, 86688170083, 124626849817, 133888707709, 143529489043, 176011837933

I could not find any solutions of 10 primes. I definitely searched all the primes up to the ones I found, so up to 176011837933.

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Emmanuel wrote:

For scheme2, my smallest  p  that gives a chain of  8  primes is  843315047  and the chain is :
843315047, 1686630137, 3373260287, 6746520587, 13493041181, 26986082369, 53972164751, 107944329529

For scheme 2, my smallest  p  that gives a chain of  8  primes is  412368577  and the chain is :
412368577, 824737189, 1649474401, 3298948813, 6597897643, 13195795309, 26391590629, 52783181291

For eventually longer chains, p  should be choosen > 2^32.

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