1. A method, comprising:receiving a message requiring public key encryption;

converting the message to a number M and generating an encrypted message E where E=Me(mod n), wherein e is at least a part of a public key and d is a private key, wherein d and e are related by de?(1 mod (totient(n))) and (e (totient(n)))=1, wherein n is a product of two predicted prime numbers computed by:

for a first index set consisting of a range of consecutive positive integers x, finding a list of prime numbers, p(x):

using prime-indexed primes (PIPs), q(s,k,x), to select values of s (prime-index-prime shift), and k (prime-index order), whereby q(s,1,x)=p(x), q(s,2,x)=p(p(x)), and so on finding a list of PIPs q(s,2,x);

performing an nth-order finite difference operation on the lists q(s,1,x) and q(s,2,x) to generate lists ?(h,n){q}(s,1,x) and ?(h,n){q}(s,2,x), where ? is a finite difference operator, h is a finite difference spacing parameter, and n is an order of finite differencing;

performing a regression of ?(h,n){q}(s,2,x) on ?(h,n){q}(s,1,x)

from the regression of ?(h,n){q}(s,2,x) on ?(h,n){q}(s,1,x), deriving a fitting function and coefficients for the fitting function;

from the fitting function and from a second index set consisting of a range of consecutive positive integers z, deriving a list of predicted finite-differenced PIPs E[?(h,n){q}(s,2,z)];

performing an inverse nth-order finite difference operation on the list E[?(h,n){q}(s,2,z)], thereby deriving a list of predicted PIPs E[q(s,2,z)];

using the derived list E[q(s,2,z)] as a means of forecasting, predicting, or refining estimated bounds of q(s,2,x); and

predicting prime numbers of at least order 1075 from the derived values of x and E[?(h,n){q}(s,2,x)] and/or E[q(s,2,x)] and;

sending the encrypted message over a network.