On the parity of exponents in the standard factorization of n! Original Research Article

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 var epsiloniset membership, variant{0,1} (i=1,2,…,k), there exist infinitely many positive integers n withimagee1(n!)≡var epsilon1(mod 2),e2(n!)≡var epsilon2(mod 2),…,ek(n!)≡var epsilonk(mod 2),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 Erdimages and Graham, that is, the conclusion with all var epsiloni=0.