Contributions towards a conjecture of Erdos on perfect powers in arithmetic progressions

Abstract

Let n, d, k ≥ 2, b, y and l ≥ 3 be positive integers with the greatest prime factor of b not exceeding k. It is proved that the equation n(n+d)... (n+d.(k−1)d) = by^l has no solution if d exceeds d₁, where d₁ equals 30 if l = 3; 950 if l = 4; 5 x 10⁴ if l = 5 or 6; 10⁸ if l = 7, 8, 9 or 10; 10¹⁵ if l ≥11. This confirms a conjecture of Erdos on the above equation for a large number of values of d.