This pedagogical note studies the elementary modular restrictions governing finite arithmetic progressions of prime numbers. For the arithmetic pattern H₍, ₊ = 0, k, 2k,. . . , (n-1) k, admissibility is shown to be equivalent to the divisibility condition P (n) | k, where P (n) is the product of all primes not exceeding n. Consequently, if p, p+k,. . . , p+ (n-1) k are primes with p>n, then the common difference must be divisible by the truncated primorial P (n), and the progression is confined to a single reduced residue class modulo P (n). The same obstruction is expressed dually for a fixed step d. If q (d) is the smallest prime not dividing d, then q (d) -1 is exactly the greatest admissible length of the associated arithmetic pattern. For the minimal admissible pattern 0, P (n), 2P (n),. . . , (n-1) P (n), the Hardy-Littlewood singular series is computed and extended to general steps P (n) m. A heuristic first-moment window scale is derived, together with an exact finite sieve-capacity bound based on the Chinese remainder theorem. The relation between finite modular survival densities and singular-series normalization is also established. Numerical experiments compare exact counts with both the elementary approximation S (n) x/ (log x) ⁿ and a more accurate shifted Hardy-Littlewood integral. The shifted model provides substantially better agreement over moderate ranges. Exact modular results and conjectural asymptotic predictions are explicitly separated throughout.
Ricardo Adonis Caraccioli Abrego (Fri,) studied this question.