On the parity of exponents in the standard factorization of n!

Let p1,p2,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given εi∈{0,1}(i=1,2,…,k), there exist infinitely many positive integers n withe1(n!)≡ε1(mod2),e2(n!)≡ε2(mod2),…,ek(n!)≡εk(mod2),where ei(n!) denotes the exponent of the prime pi in the standard factorization of positive integer n!. In 1997 Berend proved a conjecture of Erdős and Graham, that is, the conclusion with all εi=0.