The author proposes the following conjecture: Suppose $p_1, \dots, p_r$ are distinct primes and $ε_1,\dots,ε_r$ take values from $\{0,1\}$. Then there are infinitely many positive integers $n$ such that $$e_{p_1}(n!) \equivε_1 \pmod 2,\dots,e_{p_r}(n!) \equiv ε_r \pmod 2$$ where $e_{p_i}(n!)$ denotes the order of $p_i$ in $n!$. This generalizes a conjecture of Erdős and Graham which was earlier solved by {\it D. Berend} [J. Number Theory 64, 13-19 (1997; Zbl 0874.11025)]. The author proves his conjecture for the case $r=2$ and gives partial results for the general case.
Reviewer: N.Saradha (Mumbai)