Can we prove "Bertrand postulate" for primes $a \pmod q$, namely: there is always a prime number $p\equiv a \pmod q$ betwen $nq$ and $nq^2$ for every $n>0$ and $(a,q)=1$. (This would mean that betwen $nq$ and $nq^2$ there is alwajs at least $\varphi(q)$ primes, each belonging to a different residue class modulo $q$.)

Can we prove the "Bertrand postulate" for primes $a \pmod q$, namely: there is always a prime number $p\equiv a \pmod q$ betwen $nq$ and $nq^2$ for every $n>0$ and $(a,q)=1$. (This would mean that between $nq$ and $nq^2$ there are always at least $\varphi(q)$ primes, each belonging to a different residue class modulo $q$.)

Can we proove "Bertrand postulate" for primes a(mod q), namely: there is always a prime number
p=a(mod q) betwen nq and nq^2 for every n>0 and (a,q)=1. (This would mean that betwen nq and nq^2 there is alwajs at least phi(q) primes, each belong to different residuum clas mod q.)

Can we prove "Bertrand postulate" for primes $a \pmod q$, namely: there is always a prime number $p\equiv a \pmod q$ betwen $nq$ and $nq^2$ for every $n>0$ and $(a,q)=1$. (This would mean that betwen $nq$ and $nq^2$ there is alwajs at least $\varphi(q)$ primes, each belonging to a different residue class modulo $q$.)