How many primes II?

Number Theory Level 3

How many of the next list are prime numbers? a 1 = 2 + 9 ; a 2 = 2 2 + 9 ; a 3 = 2 2 2 + 9 ; ⋯ ; a n = 2 2 2 ⋯ + 9 a_1=2+9; a_2=2^2+9; a_3=2^{2^2}+9;\cdots ; a_n=2^{2^{2^{\cdots}}}+9


The answer is 2.

Gonzalo Píngaro by Gonzalo Píngaro , 45, Argentina

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1 solution

Gonzalo Píngaro
Jul 26, 2018

Notice that 2 2 2 = 16 2^{2^{2}}=16 So, 2 2 2 ⋯ = 2 4 ⋅ k = 1 6 k = 10 j + 6 2^{2^{2^{\cdots}}}=2^{4\cdot k}=16^k= 10j+6 . Then, a k + 1 = ( 1 0 j + 15 ) = 5 t a_{k+1}=(10^j+15)=5t is not a prime. Also, a 1 = 11 ; a 2 = 13 a_1=11; a_2=13 are both primes.

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