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Changes last made on Monday, 17-Dec-07 07:50:26 PST
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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