Squaring primes

 

(Dall-e creation)

Inspired by this entry of the fabulous "Futility Closet", we are proud to explore a new (?) prime challenge.

a) construct a square by stacking distinct prime numbers of the same length

b) the highest number in the stack must be a zeroless prime

c) the main descending diagonal (top left \ to bottom right) must also be a distinct prime

d) read from top of the stack to bottom, the primes involved must always be the smallest possible ones not leading to a contradiction.

We start with the 2x2 square:

2x2 earliest PFS (Prime Futility Square)

1 3

3 1

We see indeed that the 3 integers involved (13, 31 and 11) are distinct primes.

3x3 earliest PFS 

1 1 3

1 0 1

3 1 3

We see above that the 4 integers involved (113, 101, 313 and 103) are distinct primes.

4x4 earliest PFS 

1 1 1 7

1 0 0 9

1 0 1 9

7 9 9 3

5x5 earliest PFS

1 1 1 1 3

1 0 0 0 7

1 0 0 0 9

1 0 0 3 9

3 7 9 9 7 

6x6 earliest PFS

1 1 1 1 1 9
1 0 0 0 0 3
1 0 0 0 1 9
1 0 0 1 2 9
1 0 1 2 0 7
9 3 9 9 7 3

7x7 earliest PFS

1 1 1 1 1 5 1
1 0 0 0 0 0 3
1 0 0 0 0 3 3
1 0 0 0 0 3 7
1 0 0 0 0 9 9
5 0 3 3 9 8 1
1 3 3 7 9 1 1

...

As this is getting too difficult to compute by hand, I will stop my search here. I'm not even sure that the above squares are the lexicographically earliest ones :-/

If they are ok, the sequence to submit to the OEIS might be the successive horizontal "slices" of the successive k x k squares:

S = 13,31,113,101,313,1117,1009,1019,7993,11113,10007,10009,10039,37997,111119,100003,100019,100129,101207,939973,1111151,1000003,1000033,1000037,1000099,5033981,1337911,...

(All the above squares were checked and/or computed by Giorgos Kalogeropoulos – many thanks to him!)

(Dall-e creation)