Let q>r be fixed coprime positive integers. Consider the
arithmetic progression:
r, r+q, r+2q, r+3q, ...
Let R(n,q,r) be the n-th record gap between primes in this
progression.
For example, with q=6 and r=5 we have R(n,6,5)=A268928(n) in
OEIS.
Conjecture:
The n-th record prime gap R(n,q,r) almost always satisfies
the double inequality 1/6 phi(q) n^2 < R(n,q,r) < phi(q)
n^2 + (n+2)q ln^2 q
Q.
find counterexamples to the above inequalities.
