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Re: PrimePuzzles.net

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  • mgrogue <mgrogue@wi.rr.com>
    ... takes me to ... I have the same problem. I hope it is fixed soon. --Mark
    Message 1 of 4 , Feb 1, 2003
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      --- In primenumbers@yahoogroups.com, Jud McCranie <judmccr@b...> wrote:
      > I can't get onto PrimePuzzles.net (last night or this morning). It
      takes me to
      > http://www.landois.com/ instead. If I try to email Carlos Rivera, the
      > message is rejected. Are others having the same problem? Anyone know
      > what's going on?

      I have the same problem. I hope it is fixed soon.

      --Mark
    • Jens Kruse Andersen
      ... Sure you did not get your own cache? If you are sure: I know little about the Internet. Could it be an error on a regional domain name server or whatever
      Message 2 of 4 , Feb 1, 2003
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        Jose Ramón Brox wrote:
        > I just clicked http://www.primepuzzles.net/ and I didn't get any problems.

        Sure you did not get your own cache?
        If you are sure: I know little about the Internet. Could it be an error on a
        regional domain name server or whatever it's called. I am trying to access
        from Denmark. Can you e-mail Carlos and tell him about the problem others are
        having?

        Jud McCranie wrote:
        > I can't get onto PrimePuzzles.net (last night or this morning). It takes
        me to
        > http://www.landois.com/ instead. If I try to email Carlos Rivera, the
        > message is rejected. Are others having the same problem? Anyone know
        > what's going on?

        I cannot access www.primepuzzles.net either. I also get
        http://www.landois.com/ instead.
        I get a second error for www.primepuzzles.net/puzzles (browser switches
        between two messages in status line) and a third for
        www.primepuzzles.net/puzzles/puzz_209.htm (nothing happens).
        If anyone is interested in the solution for puzzle 209 which should have come
        today, my mail to Carlos is below. I don't know what the new puzzle or problem
        today should be.

        --- Begin e-mail to Carlos Rivera ---
        Puzzle 209. Triangles of primes

        1. Can you provide a formula to calculate the quantity of embedded equilateral
        triangles in an K-triangular array?

        This has to be in the EIS. A quick lookup for the first terms 1,5,13,27 finds
        www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=002717

        ID Number: A002717 (Formerly M3827 and N1569)
        Sequence: 0,1,5,13,27,48,78,118,170,235,315,411,525,658,812,988,1188,
        1413,1665,1945,2255,2596,2970,3378,3822,4303,4823,5383,5985,
        6630,7320,8056,8840,9673,10557,11493,12483,13528,14630,
        15790,17010,18291,19635,21043,22517
        Name: Floor(n(n+2)(2n+1)/8).
        Comments: Number of triangles in triangular matchstick arrangement of side n.

        Who needs to think when the EIS is there :-)

        2. Can you find one solution for every 4<K<=10?

        Each solution can be rotated and mirrored in 6 ways. I only count this as one
        solution.
        There is a single solution for K=3.
        There are 104 solutions for K=4.
        There are 1261 solutions for K=5. This is one of them:
        41
        31 37
        7 29 5
        23 17 13 19
        53 3 11 47 43

        There are no solutions for K>5.

        I first tried my search program for K=6 and found no solutions. I suspected
        there would never be more solutions. This can be proved by considering
        potentiel solutions modulo 3. The only prime which is 0 (modulo 3) is 3. The
        sum of 3 triangle vertexes must not be divisible by 3 if it has to be a prime.
        This means the modulo 3 triples (0,1,2), (1,1,1) and (2,2,2) are impossible.
        I quickly modified my existing search program to show there are no solutions
        for K>5, not even if the 0 (from 3) is omitted. This could also be proved on
        paper relatively easy.
        The above means there are no K>5 solutions for any set of distinct primes.
        There are solutions for K=6 if we allow repetitions of 3. This modulo 3
        template has two 0's, corresponding to two 3's:
        0
        2 2
        2 0 2
        1 2 2 1
        2 1 1 1 2
        1 1 2 2 1 1

        There is only one other modulo 3 template with only two 3's. That is the above
        with all 1's and 2's swapped. This would give one 2 too few to match the
        smallest primes modulo 3.
        If we replace the largest prime 73 with a second 3 then we get 214 solutions,
        all matching the above template. Here is one of them:
        3
        53 71
        11 3 23
        13 29 41 37
        59 31 67 19 47
        61 7 5 17 43 73

        For K>6 there are no template solutions and thus no prime solutions (or even
        integer solutions with prime sums) for any number of 3's allowed.

        3. Do you devise a systematic approach in order to get the solutions asked in
        2?

        I used pretty brute force for K<=6. My C program went through the vertices
        with a recursive function. It placed each allowable prime at a vertex before a
        recursive call for the next vertex. It seemed like it would be too slow for
        K=7 and I did not expect solutions. I thought of the modulo 3 simplification
        instead of optimizing the program.
        --- End e-mail to Carlos Rivera ---

        --
        Jens Kruse Andersen
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