A Prime Number Generator - Futility Closet

Take the first n prime numbers, 2, 3, 5, …, p_n, and divide them into two groups in any way whatever. Find the product of the numbers in each group, and call these A and B. (If one of the groups is empty, assign it the product 1.) No matter how the numbers are grouped, $A+B$ and $\left |A-B \right |$ will always turn out to be prime numbers, provided only that they’re less than $p_{n+1}^{2}$ (and greater than 1, of course). For example, here’s what we get for (2, 3, 5) (where $p_{n+1}^{2}$ = 7² = 49):

2 × 3 + 5 = 11
2 × 5 + 3 = 13
2 × 5 – 3 = 7
3 × 5 + 2 = 17
3 × 5 – 2 = 13
2 × 3 × 5 + 1 = 31
2 × 3 × 5 – 1 = 29

In More Mathematical Morsels (1991), Ross Honsberger writes, “For me, the fascination with this procedure seems to lie to a considerable extent in the amusement of watching it actually turn out prime numbers; I’m sure I only half believed it would work until I had seen it performed a few times.”

It makes sense if you think about it. Each of the first n prime numbers will divide either A or B but not the other, so it will fail to divide either $A+B$ or $\left |A-B \right |$ . That means that any prime divisor of $A+B$ or $\left |A-B \right |$ must be at least as big as $p_{n+1}$ , and if there were more than one of them, the number would amount to at least $p_{n+1}^{2}$ , putting it outside the limit. So for $A+B$ or $\left |A-B \right |$ between 1 and $p_{n+1}^{2}$ , it must itself be a prime number p such that p_n+1 ≤ p < $p_{n+1}^{2}$ .