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Old 2008-09-15, 12:52   #4
ckdo's Avatar
Dec 2007
Cleves, Germany

2·5·53 Posts

Originally Posted by davar55 View Post
A variation on a recent puzzle:

Draw 21 congruent circles in rows of 1,2,3,4,5 & 6, to form the shape
of an equilateral triangle.
Now fill in each circle with a different 2-digit prime (there just happen
to be 21 of these) such that the concatenation of primes in any row
of circles in either direction is prime (that's 33 primes).
Don't reverse the digits of the 2-digit primes.

Is this possible?
At the risk of being proven wrong: no.

I wrote a program to fill the triangle from the bottom up and had it dump all partial solutions with the last number (the top one) still missing. It came up with only 21 of these. However, I had previously invested some thought into pruning the search space so that no full solutions would be reported multiple times, so there may be more than just 21.

I then manually added the last number and checked the four 12-digit primes along the left and right edges for factors. In 16 cases, neither was prime. In four cases, one was prime. In one case, two were prime:

        89  23
      31  73  71
    19  67  41  13
  97  53  79  61  83
47  43  59  11  17  29
378931199747 and 479719318937 are both prime, however 372371138329 = 409 * 910442881 and 298313712337 = 116707 * 2556091.

So what I have I'd like to call a "31-prime, 35-factor" solution. A better solution in the spirit of post #2 (a) may exist, but I seriously doubt it.

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