Inequalities for inert primes and their applications

Abstract

For any given non-square integer $D\equiv 0,1(\mathrm{mod}4)$, we prove Euclid's type inequalities for the sequence $\{q_i\}$ of all primes satisfying the Kronecker symbol $(D/q_i)=-1$, $i=1,2,\cdots ,$ and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones's argument in the classification of regular ternary quadratic forms to some extent.

International Journal of Number Theory. arXiv admin note: text overlap with arXiv:1905.01423