Magic Prime-Binary-Tree

Leroy Quet qqquet at
Sat Dec 7 01:15:47 CET 2002

(Sent this to sci.math and rec.puzzles. I am posting it here because I am 
wondering about the sequence given by the number of solutions for any 
given n.)


For n = positive integer,
let us say we have n "rows", where the m_th row has
2^(m-1) "elements".
The k_th element of the m_th row is connected to the
ceiling(k/2)_th element of the (m-1)_th row (for m >= 2).

So we have a binary tree with 2^n -1 elements.

Now let each element be associated with one of every integer from 1 to
2^n - 1, assigned so that the sum of all elements (from any element in
the n_th row, to its connecting element in the (n-1)_th row, to its
connecting element in the (n-2)_th row, the single element in
row-1) is a prime.

(Not all prime sums need to be distinct.)

A n=3 example:

          /  \
       4       1
      / \     / \
     6   2   3   5

= 17,    13,  11,  13

Is there a solution for every n >= 2?
How many?

A variation: What about if we restrict the possible solutions such
that all elements of each row m, for 1 <= m <= n-1, must be summed up
to a prime as well?

(Row n will never be a prime, for it will necessarily sum 
to an even integer >= 4.)

Leroy Quet

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