Abstract :
In 1997 Berend proved a conjecture of Erdimages and Graham by showing that for every positive integer r there are infinitely many positive integers n with the property thatep(1)(n!)≡ep(2)(n!)≡…≡ep(r)(n!)≡0 mod 2,where p(1)=2, p(2)=3, p(3)=5, … is the sequence of primes in ascending order, and ep(m) denotes the order of the prime p in the prime factorization of the positive integer m. This article presents conjectures and results for the more complicated situation where an arbitrary pattern of residues modulo 2 is given.
