Problems & Puzzles: Puzzles

Puzzle 753. Prime lattices

Here we share a puzzle posted by Frank Rubin in his always interesting pages, after his kind permission.
Let s1,s2,...,sm and t1,t2,...,tn be strings of decimal digits, all of equal length. Then a prime lattice of order m,n is a set of primes
   s1t1  s1t2  ...  s1tn
   s2t1  s2t2  ...  s2tn
      ...          ...
   smt1  smt2  ...  smtn
Examples:
   11  17
   31  37
and
   1009  1051
   1109  1151
are prime lattices of order 2,2 and
   1009  1051  1087
   1109  1151  1187
   1409  1451  1487
   4909  4951  4987
is a prime lattice of order 4,3. 

Q.
Prove that there are prime lattices of every order.


Contributions was made by Antoine Verroken

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Antoine wrote:

Consider prime p escorted at a distance d by another prime p = p + d .Polya proved by a heuristic method that  R = Pd(x)/ P2(x) ,and R >= 1; Pd(x) means the number of primes that satisfy : p <= x and p + d is a prime, and P2(x) means the number of twin-primes <= x . According to Polya R = product ((p  - 1)/(p - 2) for p|d ( decomposition of d into odd prime factors ).

f.i.  d = 66  p| d = 3 , 11 or R = ( 3 1) / ( 3 2 ) * ( 11 1 ) / ( 11 2) = 2.2222

Lehmer computed up to 3.10^7 the number of primes escorted by another prime at the distance d = 2 , 4 , 6 , , 70

d                            R observed                        R theor. 

10                         1.3317                                1.3333 

30                         2.6632                                2.6667

66                         2.2186                                2.2222

70                         1.5977                                1.6000

Zhang proved that for d >= 70.000.000 the number of primes  p + d  is infinite ; with the same theory Maynard proved it for d >= 600.

Because R >= 1 probably the number of primes  p + d , d even and >=2 is infinite.

In the prime lattice of order 4.3 we see that :

               vertically             d = 100 , 300 , 3500

               horinzontally       d = 42 , 36

with the theory of Zhang and Maynard it is possible that there are prime lattices of every order.

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