A composite natural number is called primate iff:
Is it true that for all primate number, the sum of its digits is a prime number too?
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Since Primate number is composite , it has more than 2 positive divisors , and therefore has (prime) odd number of positive divisors
Remembering that for natural n > 1 , n = p 1 q 1 p 2 q 2 ⋯ p k q k , where p 1 , p 2 ⋯ p k are different primes , then , n has ( q 1 + 1 ) ( q 2 + 1 ) ⋯ ( q k + 1 ) positive divisors . So , in order to get (odd) prime number , we must have n is a (even) power of a prime number .
Considering the possible unit digit of squared number (i.e. 1 , 4 , 5 , 6 , 9 , 0 ) and notice that Primate number must have all of its digits prime , therefore , Primate number is an even power of a prime with unit digit 5 . The only such prime is 5 . Therefore , all Primate number have form 5 2 a = 2 5 a , where 2 a + 1 is prime .
Notice that for a > 1 , the last three digit of 2 5 a is 6 2 5 , and 6 is not a prime .
Therefore , the only Primate number is 2 5 . And , the sum of its digits is 7 , which is a prime number . Therefore , all Primate number have prime digital sum .