mersenneforum.org What is the currently largest known bi-twin primes?
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 2021-08-28, 10:42 #1 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3×19×53 Posts 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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