Arithmetic Progressions with Common Difference Divisible by Small Primes

n(n + d) . . . (n + (k − 1)d) = by (1.1) in positive integers n, k ≥ 2, d > 1, b, y, l ≥ 3 with l prime, gcd(n, d) = 1 and P (b) ≤ k. We write d = D1D2 (1.2) where D1 is the maximal divisor of d such that all prime divisors of D1 are congruent to 1 ( mod l). Thus D1 and D2 are relatively prime positive integers such that D2 has no prime divisor which is congruent to 1 (mod l). Shorey [Sh88] proved that (1.1) implies that