# Partitions of Primes

Problem 461 from Parabola (Volume 16, Issue 2, p.32) asked: Partition the set $P_n=\{2,3,5,\dots,p_n\}$ of the first $n$ primes into two nonempty disjoint parts $A,B$ and let $a, b$  be their respective products. Is $|a-b|$ always a prime or 1? If not, find the smallest $n$ for which it isn't. The answer (given in Parabola 1981, volume 17, issue 1, p.~31--32) is  no, and the smallest $n$ is 5. Taking $A=\{2,5,7,11\}$ and $B=\{3\}$, one has $a=770, b=3$ and $a-b=767=13\cdot59$. To see this, the key observation is that the numbers $a,b$ share no common factors. It follows that the  prime factors of $a-b$  can't divide either $a$ or $b$. So the smallest possible prime factor of  $a-b$ is $p_{n+1}$. Armed with this information, it doesn't take long to find the required answer. And this is all done easily by hand; after all, the problem was posed in 1980. We propose a modern variation of this problem.