Problems & Puzzles: Puzzles

Puzzle 682 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 r2,...,rd fixed".

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 progression"

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 solution to
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 arithmetic
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 following conditions:

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 (either
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 row.

Q1. Find the arrays satisfying conditions 1), 2), 3) above for
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 disjoint).

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.



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