In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆ { 1 , ⋯ , n } in k-sets such that the first $$k-1$$ k - 1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial $$f_A(x) := \prod _{x \in A} (x-a)$$ f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation $$f_A(x)=P(y)$$ f A ( x ) = P ( y ) where $$P(y)\in \mathbb {Q}[y]$$ P ( y ) ∈ Q [ y ] and on shifted power values of $$f_A(x)$$ f A ( x ) .

We give sharp, in some sense uniform bounds for the number of $$\ell $$ -th powers and arbitrary powers among the first N terms of an arithmetic progression, for N large enough.

Products of terms of arithmetic progressions yielding a perfect power have been long investigated by many mathematicians. In the particular case of consecutive integers, various finiteness results are known for the polynomial values of such products. In the present paper we consider generalizations of these result in various directions.

We investigate certain arithmetic properties of factorials. On the one hand, we are interested in the densities of sets of $n$ such that the exponents of given primes in the prime factorization of $n!$ hold certain congruence properties. On the other hand

We consider the equation in the title in positive integers A,B,C. We give an explicit upper bound for C in terms of the difference k:=B−A. Further, we show that for k≤106 this equation has only one (long known) non-trivial solution, given by 6!7!=10!.