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Old 2021-08-28, 10:42   #1
sweety439
 
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Default What is the currently largest known bi-twin primes?

What is the currently largest known bi-twin primes? i.e. n +/- 1 and 2*n +/- 1 all primes? These 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 twin primes
* 2*n +/- 1 are twin primes
* n-1 and 2*n-1 are Sophie Germain primes and safe primes of the first kind or Cunningham chain of the first kind.
* n+1 and 2*n+1 are Sophie Germain primes and safe primes of the second kind or Cunningham chain 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:

https://www.primepuzzles.net/problems/prob_049.htm
https://www.rieselprime.de/Related/RieselTwinSG.htm
https://harvey563.tripod.com/cunninghams.txt

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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