PRIMAL SCREAM

Counting RSA-integers

In the RSA cryptosystem integers of the form n=p.q with p and q primes of comparable size (`RSA-integers') play an important role.

It is a folklore result of cryptographers that C_r(x), the number of integers n<=x that are of the form n=pq with p and q primes such that p<q<rp, is for fixed r>1 asymptotically equal to c_r*x*log^{-2}x for some constant c_r>0.

Here we prove this and show that c_r=2log r.

About the logic of the prime number distribution

There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5
and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1.

All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3.