Recursive prime brother by Brillhart – Lehmer – Selfridge algorithm

Define:

p[k,i]=ABS[1+2*n[k,i]*p[k-1,1]*p[k-1,2]],n[k,1] is the integer with minimum ABS[n[k,1]] that makes p[k,1] a prime number, and n[k,2] is the integer with second minimum ABS[n[k,2]] that makes p[k,2] a prime number

The primality of p[k,i] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[k-1,1] and p[k-1,2] as helper since n is a small integer, by reducing k to 1.

With this idea, taking

p[0,1]=1, p[0,2]=1
We got the n[i,j] ( columns : i; rows: k):

The last three, p[19,1] makes top 5000 list.
The proof will be posted in the reply of this one.