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FOUR OR LESS (Posted on 2010-07-05) Difficulty: 2 of 5
Determine all prime numbers p such that the total number of positive divisors of A = p^2 + 1007 (including 1 and A) is 4 or less.

  Submitted by Ady TZIDON    
Rating: 4.0000 (1 votes)
Solution: (Hide)
ans: P=2 SOLUTION:

Prime number p is either odd or 2.
Odd squared number equals 1 modulo 4, and 1007 is -1 mod 4, so A is divisible by 4.
This means it has at least 6 divisors: 1, 2, 4, A/4, A/2, A.

If p equals 2, 1011 (=4+1007=3*337), has exactly 4 factors, 1, 3, 337, 3*337.

Thus, p=2 is the only valid answer.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionLooking at pGamer2010-07-05 14:34:43
re: an analytical solution (at least an attempt)Steve Herman2010-07-05 14:10:11
Solutionan analytical solution (at least an attempt)Dej Mar2010-07-05 14:02:16
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