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Old 2020-10-24, 16:20   #1
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Nov 2016

9C016 Posts
Default 47 & 383

They are not only two primes of the form 3*2^n-1, but also ....

* They are the first two non-Sophie-Germain primes (i.e. primes p such that 2*p+1 is not prime) of the form 3*2^n-1

* They are the largest two numbers n such that the primary pretender of n (A000790) = n-1

* They are the largest two primes p such that the smallest pseudoprime base p (A090086) >= p-1

* They are both indices of the records for the smallest k such that n*2^k+1 is prime (47 gets 583, 383 gets 6393, both are >10 times of the previous record)

* They also have no easy prime of the form either 2*n^k+1 or n^k+2 (these two forms are dual) (for 47, they are 2*47^175+1 and 47^113+2, and there are no (probable) prime for both two forms for 383)
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