SQUARES IN PRODUCTS IN ARITHMETIC PROGRESSION WITH AT MOST ONE TERM OMITTED AND COMMON DIFFERENCE A PRIME POWER SHANTA LAISHRAM, T. N. SHOREY, AND SZABOLCS TENGELY Abstract. It is shown that a product of k − 1 terms out of k ≥ 7 terms in arithmetic progression with common difference a prime power > 1 is not a square. In fact it is not of the form by 2 where the greatest prime factor of b is less than or equal to k. Also, we show that product of 11 or more terms in an arithmetic progression with common difference a prime power > 1 is not of the form by 2 where the greatest prime factor of b is less than or equal to p π(k)+2 . 1. Introduction For an integer x> 1, we denote by P (x) and ω(x) the greatest prime factor of x and the number of distinct prime divisors of x, respectively. Further we put P (1) = 1 and ω(1) = 0. Let p i be the i−th prime number. Let k ≥ 4,t ≥ k − 2 and γ 1 <γ 2 < ··· <γ t be integers with 0 ≤ γ i <k for 1 ≤ i ≤ t. Thus t ∈{k,k − 1,k − 2},γ t ≥ k − 3 and γ i = i − 1 for 1 ≤ i ≤ t if t = k. We put ψ = k − t. Let b be a positive squarefree integer and we shall always assume, unless otherwise specified, that P (b) ≤ k. We consider the equation (1.1) Δ = Δ(n,d,k)=(n + γ 1 d) ··· (n + γ t d)= by 2 in positive integers n,d,k,b,y,t. It has been proved (see [SaSh03a] and [MuSh04a]) that (1.1) with ψ =1,k ≥ 9, d ∤ n, P (b) <k and ω(d) = 1 does not hold. Further it has been shown in [TSH06] that the assertion continues to be valid for 6 ≤ k ≤ 8 provided b = 1. We show Theorem 1. Let ψ =1,k ≥ 7 and d ∤ n. Then (1.1) with ω(d)=1 does not hold. Thus the assumption P (b) <k and k ≥ 9 (in [SaSh03a] and [MuSh04a]) has been relaxed to P (b) ≤ k and k ≥ 7, respectively, in Theorem 1. As an immediate consequence of Theorem 1, we see that (1.1) with ψ =0,k ≥ 7, d ∤ n, P (b) ≤ p π(k)+1 and ω(d) = 1 is not possible. If k ≥ 11, we relax the assumption P (b) ≤ p π(k)+1 to P (b) ≤ p π(k)+2 in the next result. Theorem 2. Let ψ =0,k ≥ 11 and d ∤ n. Assume that P (b) ≤ p π(k)+2 Then (1.1) with ω(d)=1 does not hold. For related results on (1.1), we refer to [LaSh07]. AMS Classification: Primary 11D61; Keywords: Diophantine equations, Arithmetic Progressions, Le- gendre symbol. Research of S. Tengely was supported in part by the Magyary Zolt´ an Higher Educational Public Foundation. 1