[seqfan] Prime partionable numbers and {P1} and {P2}.

[Chris]

Dear Seqfans,

 

W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr.
Math. 22 (1978) 263-272,

Definition 5.3 on Page 269, defines a prime-partitionable number as follows:

"An integer n>=2 is said to be prime-partitionable if there is a partition
{P1,P2} of the set P of all

primes less than n such that, for all natural numbers n1 and n2 satisfying
n1+n2=n, we have

(gcd (nl, pl), gcd (n2, p2)) does not equal (1,1) for some (p1,p2) belonging
to PI X P2."

 

The first prime-partitionable number is 16. The set of primes < 16 = {2, 3,
5, 7, 11, 13}.

There are two partitions of P that demonstrate that 16 is
prime-partitionable, denoted by Pt1 and Pt2:

  Pt1: P1 = {2, 5, 11},    P2 = {3, 7, 13}

  Pt2: P1 = {2, 3, 7, 13}, P2 = {5, 11}

                   Pt1         Pt2

       n1   n2   p1   p2     p1   p2 

        1 + 15    -    3      -    5

        2 + 14    2    7      2    -

        3 + 13    -   13      3    -

        4 + 12    2    3      2    -

        5 + 11    5    -      -   11

        6 + 10    2    -      2    5

        7 +  9    -    3      7    -

        8 +  8    2    -      2    -

        9 +  7    -    7      3    -

       10 +  6    2    3      2    -

       11 +  5   11    -      -    5

       12 +  4    2    -      2    -

       13 +  3    -    3     13    -

       14 +  2    2    -      2    -

       15 +  1    5    -      3    -

For each n1+n2, either p1|n1 or p2|n2 or both.

 

I have determined all partitions of P that demonstrate that n is
prime-partitionable for n in

{16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96} and have
noticed that each P1 and

each P2 contains one or more prime partitions of n, e.g. in P1 of Pt1,
5+11=16; in P2 of Pt1, 3+13=16;

in P1 of Pt2, 3+13=16; in P2 of Pt2, 5+11=16.

 

Is this generally true? Can someone explain it?

 

Best regards,

Chris Gribble