Problems & Puzzles:
Stanley Antimagic Squares-II
This is a follow up to
Puzzle 681 based in the
contribution that J. Wroblewski made to it.
Remember that his contribution disregarded the
minimal condition asked in Puzzle 681 to the index (sum) S of
the Stanley Antimagic Squares and asks for solutions with the
largest dimension d possible. He stated that we may get a
solution if we are able to find "d disjoint
configurations of d primes each. By a configuration I mean a set of d
primes with predefined differences, i.e. p, p+r2, p+r3, ..., p+rd with
Moreover he suggested that a bigger solution
than the gotten for him for Puzzle 681 (d=20) could emerge if we
could find "more frequent configuration of primes instead an arithmetic
Now, let's show directly the puzzle that Wroblewski
has posed for this issue:
We are interested in prime constellations of k primes with
predefined differences, i.e. prime sequences of the form
(p,p+d2,p+d3,...,p+dk), where d2<d3<...<dk are fixed
positive integers. We would like to know what is the shape
of constellations that appear most frequently at a certain
range for a given k. Are those arihtmetic progressions? If
so, should they have a primorial difference?
The ultimate goal is to work out a strategy of finding a
Puzzle 681 with the index minimality condition 2) dropped,
but with the size d as large as possible. An example for
d=20 can be
constructed by using 20 prime constellations being
progressions with the common difference of 43#. Could we do
better searching for some constellations other than
arithmetic progressions instead?
To be more strict in formulation, for a given k=2,3,4,...
and given n much larger than k, we would like to find an
array of primes, with k columns and n rows, satisfying the
1) Primes in any row or column form an increasing sequence
2) Denoting by p(i,j) the prime in i-th row and j-th column,
we have p(i1,j1)+p(i2,j2)=p(i1,j2)+p(i2,j1)
for any 1<=i1<i2<=n and
3) The largest prime in the array is as small as possible
proven minimal or the smallest we can find)
Note that the condition 2) is equivalent to saying that
examples of the same constellation are being listed row by
Q1. Find the arrays satisfying conditions 1), 2), 3) above
k=2,3,4,5 and 20<=n<=1000. Observe how the first row changes
when n goes from 20 to 1000. That comes down to observing a
race between various constellation.
Q2. Redo Q1 with the additional condition: all the primes in
the array must be distinct (i.e. constellations must be
Q3. Redo Q1 and Q2 for k=6,7,8,9,10.
Q4. Redo Q1, Q2 and Q3 with the additional condition that
rows contain arithmetic progressions.