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Not every prime qualifies! (Posted on 2017-05-15) Difficulty: 2 of 5
Find all primes p such that 11+p^2 has exactly six different positive divisors (the number itself included).

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution | Comment 1 of 2
The only prime that qualifies is 3.

All primes except 2 and 3 can be written as (6n+1) or (6n+5) for positive integer n.
(2 does not qualify.  11+2^2=15=3*5 has only 4 divisors.)

11+(6n+1)^2=36n^2+12n+12=12(n^2+n+1) which has a minimum of 12 divisors if (n^2+n+1) is prime.

11+(6n+5)^2=36n^2+60n+36=12(3n^2+5n+3) which similarly has a minimum of 12 divisors.

  Posted by Jer on 2017-05-15 09:25:46
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