3 fixed title

# Find the least prime with(atleast)sothatp-1has two factors greater than $m$ and $n$

Given positive integers $m$ and $n$, what is an explicit upper bound on the least prime $p$ such that $p-1$ has factors $m+i$ and $n+j$ for some $i,j \geq 0$? In other words, some number at least as large as $m$ divides $p-1$, and so does some number at least as large as $n$.

I would also appreciate any pointers to related number-theoretic conjectures. Thanks very much

Example:

Suppose $m=5$ and $n=4$. Then, while 23 is prime, and $23 > 20$, it does not qualify because 22 can only be factored into 11 and 2, and $2< m,n$. On the other hand, 29 is ok, because 28 can be factored into 7 and 4, both of which are at least as large as $m,n$.

1

# Find the least prime with (at least) two factors greater than $m$ and $n$

Given positive integers $m$ and $n$, what is an explicit upper bound on the least prime $p$ such that $p-1$ has factors $m+i$ and $n+j$ for $i,j \geq 0$? In other words, some number at least as large as $m$ divides $p-1$, and so does some number at least as large as $n$.