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 sweety439 2021-08-28 10:42

What is the currently largest known bi-twin primes?

What is the currently largest known [URL="http://web.archive.org/web/20061015181556/http://ourworld.compuserve.com/homepages/hlifchitz/Henri/us/NouvChPus.htm"]bi-twin primes[/URL]? i.e. n +/- 1 and 2*n +/- 1 all primes? [URL="http://web.archive.org/web/20061015181757/http://ourworld.compuserve.com/homepages/hlifchitz/Henri/fr-us/BiTwinRec.htm"]These[/URL] are the records in 2006, but this list has not been updated for a long time.

If n +/- 1 and 2*n +/- 1 are all primes, then:

* n +/- 1 are [URL="https://en.wikipedia.org/wiki/Twin_prime"]twin primes[/URL]
* 2*n +/- 1 are [URL="https://en.wikipedia.org/wiki/Twin_prime"]twin primes[/URL]
* n-1 and 2*n-1 are [URL="https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes"]Sophie Germain primes and safe primes[/URL] of the first kind or [URL="https://en.wikipedia.org/wiki/Cunningham_chain"]Cunningham chain[/URL] of the first kind.
* n+1 and 2*n+1 are [URL="https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes"]Sophie Germain primes and safe primes[/URL] of the second kind or [URL="https://en.wikipedia.org/wiki/Cunningham_chain"]Cunningham chain[/URL] of the second kind.

Also a problem: Find and proof the smallest k divisible by 15 such that k*2^n +/- 1 and k*2^(n+1) +/- 1 cannot be prime simultaneously for all integers n>=1? (such k must divisible by 15 since if k is not divisible by 3, then one of n +/- 1 (also one of 2*n +/- 1, one of n-1 and 2*n-1, one of n+1 and 2*n+1) will be divisible by 3, and if k is not divisible by 5, then one of n +/- 1 and 2*n +/- 1 will be divisible by 5)

References of similar problems:

[URL="https://www.primepuzzles.net/problems/prob_049.htm"]https://www.primepuzzles.net/problems/prob_049.htm[/URL]
[URL="https://www.rieselprime.de/Related/RieselTwinSG.htm"]https://www.rieselprime.de/Related/RieselTwinSG.htm[/URL]
[URL="https://harvey563.tripod.com/cunninghams.txt"]https://harvey563.tripod.com/cunninghams.txt[/URL]

and the conjectured smallest k:

* k*2^n +/- 1: k = 237
* k*2^n-1 and k*2^(n+1)-1: k = 807
* k*2^n+1 and k*2^(n+1)+1: k = 32469

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