Golygon Letter to Mr. A. K. Dewdney




        July 15, 1990

        19628 Via Monte Drive
        Saratoga, CA 95070

        Mr. A. K. Dewdney, Mathematical Recreations
        c/o The Editors, Scientific American
        415 Madison Avenue
        New York, NY 10017

        Enclosures: Sample of computer printout of Golygon program.

        Dear Mr. Dewdney:

        Your  discussion  of golygons in Scientific American,  July  1990 
        caused me to write a computer program to allow me to  investigate 
        some "new avenues of inquiry".

        Your  question  "Can we find prime-sided  golygons?  ...  of  odd 
        primes:  1,  3,  5,  11, 13 and so on."  was my first inquiry.  I 
        found  two  solutions,  both  of which  are  16-sided.   The  two 
        solutions are: 

        G16,P1,3,105 = 1N 3E 5N 7W 11N 13W 17N 19E 23N 29W 31N 37E 
                       41S 43E 47S 53W

        and

        G16,P1,102,105 = 1N 3E 5S 7W 11S 13W 17N 19E 23N 29W 31S 37E
                         41S 43E 47N 53W.

        In  the  notation  G16,P1,3,105  the G16 stands  for  a  16-sided 
        golygon,  the  P1 stands for a sequence of consecutive odd primes 
        starting  with  1,  the  3 is the 'summing type' number  for  the 
        north-south  directions and the 105 is the 'summing type'  number 
        for  the east-west directions.   If you convert the summing  type 
        number 105 to 8-bit binary you get 01101001, now convert 0's to + 
        and 1 to -.   Since this is the east-west summing type number, we 
        have E W W E W E E W.

        I see no reason to require the first side of a golygon to  always 
        be  equal to one.   If you follow the new avenue of allowing  the 
        first  side  to be greater than one,  many more golygons  can  be 
        found, for example:

        G8,P359,6,6      G8,P389,6,6       G8,P839,6,6

        G12,P97,25,25    G12,P313,26,26    G12,P1013,25,22

        G16,P3,105,37    G16,P3,105,67     G16,P17,13,90
        G16,P19,90,102   G16,P1069,58,58   G16,P1381,113,106.

        If  we  turn  our  attention  back  to  the  consecutive  integer 
        golygons,  but  with not requiring the first side to be  one,  we 
        find:

        G8,I1,6,6 = 1N 2E 3S 4W 5S 6W 7N 8E
        G8,I2,6,6 = 2N 3E 4S 5W 6S 7W 8N 9E
        G8,I3,6,6 = 3N 4E 5S 6W 7S 8W 9N 10E
          . . .

        This  goes  on forever since the summing type is + - - + and  has 
        the same number of +'s and -'s.  We will not count a golygon that 
        is  generated  by adding one to all sides with the  same  summing 
        types  as  a  new golygon.   We can still find some  new  golygon 
        though, for example:

        G16,I1,60,14     G16,I1,60,21     G16,I1,60,35     G16,I1,60,60  
        G16,I1,60,90     G16,I1,60,102    G16,I1,60,105    G16,I1,90,14  
        G16,I1,90,21     G16,I1,90,35     G16,I1,90,60     G16,I1,90,90  
        G16,I1,90,102    G16,I1,90,105    G16,I1,102,14    G16,I1,102,21  
        G16,I1,102,35    G16,I1,102,60    G16,I1,102,90    G16,I1,102,102  
        G16,I1,102,105   G16,I1,105,14    G16,I1,105,21    G16,I1,105,35  
        G16,I1,105,60    G16,I1,105,90    G16,I1,105,102   G16,I1,105,105  

        G16,I2,14,60     G16,I2,14,90     G16,I2,14,102    G16,I2,14,105  
        G16,I2,21,60     G16,I2,21,90     G16,I2,21,102    G16,I2,21,105  
        G16,I2,35,60     G16,I2,35,90     G16,I2,35,102    G16,I2,35,105  
        G16,I3,60,13     G16,I3,60,19     G16,I3,90,13     G16,I3,90,19  
        G16,I3,102,13    G16,I3,102,19    G16,I3,105,13    G16,I3,105,19  
        G16,I4,13,60     G16,I4,13,90     G16,I4,13,102    G16,I4,13,105  
        G16,I4,19,60     G16,I4,19,90     G16,I4,19,102    G16,I4,19,105  
        G16,I5,60,11     G16,I5,90,11     G16,I5,102,11    G16,I5,105,11  
        G16,I6,11,60     G16,I6,11,90     G16,I6,11,102    G16,I6,11,105  
        G16,I7,60,7      G16,I7,90,7      G16,I7,102,7     G16,I7,105,7  
        G16,I8,7,60      G16,I8,7,90      G16,I8,7,102     G16,I8,7,105  

        These are all of the 16-sided golygons there are, all 72 of them.  
        The first 28 are the same as the ones you published, the rest are 
        due to allowing the first side to be greater than one.   When the 
        first  side is nine or larger,  no new 16-sided golygons  can  be 
        found.

        My  last  new avenue of inquiry was:  Can we find golygons  whose 
        consecutive  sides are consecutive twin primes that do  not  have 
        non-twin primes between them?   This was prompted by some earlier 
        investigation  I  did  with twin  primes.   Two  consecutive  odd 
        numbers  that are both prime are twin primes.   I defined a  quad 
        prime  to  be four consecutive primes that are two sets  of  twin 
        primes.   A sextet prime is six consecutive primes that are three 
        sets of twin primes.   An octet prime is eight consecutive primes 
        that are four sets of twin primes.  Some of the ones that I found 
        are:

        G8,T663569,6,6 = 663569N 663571E 663581S 663583W
                         663587S 663589W 663599N 663601E
        G8,T909287,6,6     G8,T1006301,6,6    G8,T1159187,6,6
        G8,T2502341,6,6    G8,T3664679,6,6    G8,T7129217,6,6
        G8,T10187909,6,6   G8,T10531061,6,6   G8,T11495579,6,6
        G8,T12628337,6,6   G8,T13225997,6,6   G8,T14327639,6,6
        G8,T16288199,6,6.

        These  are  all  14 of the 8-sided golygons made  of  Octet  twin 
        primes less than 2 to the 24th power.


        Sincerely,



        Harry J. Smith


Notes: Sequences have been defined at The On-Line Encyclopedia of Integer Sequences! for
A007530 Prime quadruples: numbers n such that n, n+2, n+6, n+8 are all prime.
A136141 First of four consecutive primes that are two sets of twin primes.
A136142 Four consecutive primes with two sets of twin primes.
A136143 First of six consecutive primes that are three sets of twin primes.
A136144 Six consecutive primes with three sets of twin primes.
A136145 First of eight consecutive primes that are four sets of twin primes.
A136146 Eight consecutive primes with four sets of twin primes.

Go to sequence A136146 Eight consecutive primes with four sets of twin primes.

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