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 à 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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