On the parity of exponents in the standard factorization of \(n!\) (Q1394928)

Let \(2=p_1<p_2<\dots<p_i<\dots\) be the sequence of primes and denote by \(e_i(m)\) the exponent of the prime \(p_i\) in the standard factorization of the positive integer \(m\). \textit{D. Berend} [J. Number Theory 64, 13--19 (1997; Zbl 0874.11025)] proved a conjecture of Erdős and Graham proving that for every \(k\) there are infinitely many integers \(n\) for which \(e_i(n!)\) an even number for every \(i=1,2,\dots k\). In the present paper the author proves a generalization of this theorem showing: for every integer \(k\) and the sequence \(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_k\) of 0s and 1s (i.e. \(\varepsilon_i\in\{0,1\}\)) there are infinitely many integers \(n\) for which \(e_i(n!)\equiv \varepsilon_i\pmod 2\), \(i=1,2,\dots, k\).