Find the number of natural numbers for which the infinitely nested radical expression above is a prime number?
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Let f ( n ) = n + n + n + n + ⋯ f 2 ( n ) = n + f ( n ) f 2 ( n ) − f ( n ) − n = 0 f ( n ) = 2 1 + 2 1 + 4 n -ve is neglected since f(n) is +ve for f(n) to be a prime number , Let 1 + 4 n = m 2 f ( n ) = 2 1 ( 1 + m ) 1 + m must be an even no. , Let 2k f ( n ) = k now , n = 4 1 [ ( 2 k − 1 ) 2 − 1 ] n = k 2 − k ≤ 2 0 1 6 where k is prime Possible values of k = ( 2 , 3 , 5 , 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 , 3 1 , 3 7 , 4 1 , 4 3 ) So our answer is 1 4