A051252 - Prime Circles - Summary and Update

[T. D. Noe]

Nice summary.

>As Mike Hennebry pointed out, if 2m+1 and 2m+3 are both prime, then we
>can definitely find a hamilton cycle in G(2m). For example, this is the
>case with G(10) where 11 and 13 are both prime. We find the Hamilton
>cycle as follows:

Mike's technique can be generalized.  For many m, it is possible to rotate
the even numbers so that a solution is found.  Just before I received this
e-mail, I submitted three sequences related to this.  What a coincidence!
They are

A072616

1,1,1,1,4,8,2,5,18,2,9,100,80,224

Number of essentially different ways of arranging numbers 1 through 2n
around a circle so that the sum of each pair of adjacent numbers is prime
and the odd (or even) numbers are in order.

A restricted form of the prime circle problem whose sequence is A051252.
Note that a(2)=1 because the two solutions are essentially the same.  The
number of solutions is the same for odd or even numbers in order because a
solution having the odd numbers in order can be converted to a solution
having even numbers in order by subtracting 1 from even numbers and adding
1 to odd numbers.  For example, {1,2,3,8,5,6,7,4,9,10} becomes
{2,1,4,7,6,5,8,3,10,9}. Is the number of solutions always positive?  See
A072617 for some simple solutions to the prime circle problem.

A072617

1,1,1,1,1,2,1,1,1,0,0,2,1,1,2,0,0,1,1,1,3,0,0,1,0,0,2,1,1,2,0,0,2,1,1,1,0,0,3,
0,0,1,0,0,3,0,0,3,1,1,1,1,1,3,0,0,0,0,0,5,0,0,3,0,0,4,1,1,4,0,0,2,1,1,2,0,0,2,
0,0,4,0,0,5,0,0,4,1,1,5,0,0,3,1,1,2,1,1,4,0

Number of essentially different ways of arranging numbers 1 through 2n
around a circle so that the sum of each pair of adjacent numbers is prime,
with the odd and even numbers in order in opposite directions.

A very restricted form of the prime circle problem whose sequence is
A051252.  Finding these solutions is very fast because there are only n
possible solutions to try.  See A072616 for the case where only the odd
numbers or only the even numbers are in order.  Note that a(2)=1 because
the two solutions are essentially the same.  Solutions can be printed by
removing comments from the Mathematica program.  Is there a simple rule
that determines when n has a positive number of solutions?

A072618

1,2,3,4,5,6,7,8,9,12,13,14,15,18,19,20,21,24,27,28,29,30,33,34,35,36,39,42,45,
48,49,50,51,52,53,54,60,63,66,67,68,69,72,73,74,75,78,81,84,87,88,89,90,93,94,
95,96,97,98,99

n for which the prime circle problem has a simple solution: the arrangement
of numbers 1 through 2n around a circle is such that the sum of each pair
of adjacent numbers is prime, and the odd and even numbers are in order in
opposite directions.

A very restricted form of the prime circle problem whose sequence is
A051252.  This sequence lists the n for which A072617(n) is positive.  See
A072616 for the case where only the odd numbers or only the even numbers
are in order.  Is there a simple rule that determines when n is on this
list?


The submissions include Mathematica code.

Best regards,

Tony