Let $p$ be prime and $p{\#}$ the product of all primes not larger than $p$. Are there any positive integers $x$ and $y$ such that $pxy+x+y=p{\#}$. It appears there are no solutions. There are no solutions with $p<200$. Can it be shown that this is the case for all primes $p$?
My thoughts: Let $n=pxy+x+y$. If we can show that if $q \ | \ n$ for all primes $q \le p$ and $q\nmid n$ for all primes $q>p$ then $q^2 \ | \ n$ for some prime $q \le p$, then we are done. (Note: The proposition is false for higher powers of $p{\#}$. It's also false when $p{\#}$ is replaced by $p!$ , or $p{\#} + b, b \not = 0$)