Do there exist three 2-digit primes such that:

• Any two of the three, averaged, produce another prime, and
• The average of all three is prime

Here’s how I approached it… it was sort of elimination, though could be called brute force (though not really in my opinion). First I listed all the 2 digit primes (there’s 21 of them). Then I multiplied each by 2. Why? Because the average of any pair of primes is (P1 + P2)/2. In order for the average to be prime, the (P1 + P2) sum MUST be some prime times 2.

So now I have a list of the primes from 11-97, and I have a list of primes*2 from 22-194.

Next, I started with 11 and added it to each of the primes that are GREATER than 11 (and less than 97 of course). If the sum was in the list of primes*2, then I made a note of the prime that had been added to 11. Then I continued to 13, and added it to each of the primes that are GREATER than 13, and repeated the process. So here is what I got:

11: 23, 47, 71, 83
13: 61, 73
17: 29, 41, 89
19: 43, 67
23: 59, 71, 83
29: 53, 89
31: 43
37: 97
41: 53
43: 79
47: 59, 71
53: 89
59: 83
61: 73, 97
67: 79

I stopped listing at 67 because the primes greater than 67 didn’t have any primes GREATER than itself that fit my criteria.

Well, I am going to cross out (but not ignore) all the rows that only have two primes total (so the header prime and only one in the list). If they are part of a solution, it will be found in one of the lower prime’s list (you’ll see what I mean in a second).

Next, I can quickly look at the 4 rows that have 3 primes total. For example, 13: 61, 73. I could look down my list to see if the other two numbers are listed together (like "is 73 in 61’s list?) or I could just take the average of the three and see if it’s prime. Well, I’m going to have to check all 3 together eventually, so I’ll just do that first. Only 1 of these 4 groups results in a prime when you average the three: 47, 59, 71. However, notice that 71 is not in 59’s list. So this is not a solution. So I can cross out (but not ignore) the 4 rows that have 3 primes total.

So the only rows I should really be paying attention to are the ones for 11, 17, and 23. Like I said, I’ll be referencing the other rows to see if I am dealing with reasonable trios.

Ok, well with 11, I can look and see if 47, 71 or 83 appear in 23’s list. Well 71 and 83 do. So (11, 23, 71) and (11, 23, 83) are reasonable trios. Next, I can look and see if 71 or 83 appear in 47’s list. 71 does, so (11, 47, 71) is a reasonable trio. The next step would be to see if 83 appears in 71’s list (to see if 11, 71, 83 is a reasonable trio), but as 71 doesn’t have a list, this means there are no primes greater than it that meet my previous criteria. So I’m done with 11’s list.

I can do the same thing for 17 and see that only (17, 29, 89) is a reasonable trio, and for 23 only (23, 59, 83)

So of all the combinations I might have tried to "brute force" check, I am only checking 5 groups. And the only one where the average of all 3 is still a prime is 11, 47, 71.

Posted by nikki on 2004-10-22 14:29:34