On the products (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1), II

TextIn this paper, the following results are proved: (i) For any odd integer ℓ with at most two distinct prime factors and any positive integer n  , the product (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1) is not a powerful number; (ii) For any integer r≥1r≥1, there exists a positive integer TrTr such that, if ℓ is a positive odd integer with at most r distinct prime factors and n   is an integer with n≥Trn≥Tr, then (1ℓ+1)(2ℓ+1)⋯(nℓ+1)(1ℓ+1)(2ℓ+1)⋯(nℓ+1) is not a powerful number.VideoFor a video summary of this paper, please visit http://youtu.be/nU-nkxNX1BA.