On the parity of exponents in the factorization of \(n\)! (Q1356233)

Let \(2=p_1<p_2<\cdots\) be the sequence of all primes in ascending order. Furthermore, let \(n!= p^{\alpha_1(n)}_1\cdots p^{\alpha_m(n)}_m\). \textit{P. Erdös} and \textit{R. Graham} in their celebrated problem book [Old and new problems and results in combinatorial number theory, Monographie No. 28 de L'Enseign. Math. (1980; Zbl 0434.10001)] raised the following question: Does there exist, for every \(k\), some \(n>1\) with all exponents \(\alpha_1(n),\alpha_2(n),\dots,\alpha_k(n)\) even? In the present paper, the author gives an affirmative answer to this question. His proof is elementary. Furthermore, the author proves that for arbitrary fixed positive integers \(m\), \(k\), \(d\) there exist infinitely many positive integers \(n\) for which all the exponents \(\alpha_1(tn),\alpha_2(tn),\dots,\alpha_k(tn)\), \(1\leq t\leq m\), are divisible by \(d\). The proof of this theorem uses some results of dynamical systems.