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.

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