Bertrand's Postulate for Primes in Arithmetical Progressions

Abstract Bertrand's Postulate is the theorem that the interval ( x , 2 x ) contains at least one prime for x \2>1. We prove, building on work of Erd\”os, analogues of this result, in which the interval is of the form ( x, zx ) and there are at least m primes ≡ a (mod d ) required to be contained in this interval, and where z, a and d have to satisfy some conditions. For the case m  1 the results are worked out using a computer. They can be found in Table 1.