Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Let P(n)= The product of the first n primes.

P(n)+1 is like Euclid's method to show there are infinitely many primes, and may or not be prime itself. Now look at the difference between P(n) and the *next* prime after P(n)+1.

n=1, 5-(2)=3

n=2, 11-(2x3)=5

n=3, 37-(2x3x5)=7

n=4, 223-(2x3x5x7)=13

n=5, 2333-(2x3x5x7x11)=23

n=6, 30047-(2x3x5x7x11x13)=17

n=7, 510529-(2x3x5x7x11x13x17)=19

n=8, 9699713-(2x3x5x7x11x13x17x19)=23

Are these differences always prime?

The Dinner and Dialogue Club has planned a series of small meetings. Each meeting would consist of two or three members enjoying friendly conversation with each other while eating food from different places all over the world. Each member is scheduled to meet exactly four times. No pair of members will meet twice, but some pairs might not meet at all.

The first thing the club did was schedule and arrange the meetings so that each member knew whom to meet and when. When it came to choosing restaurants, someone suggested that each member eat at two restaurants with eastern food, and two with western food (each restaurant is either one or the other). They liked the idea, but to their dismay, the idea was not possible without rearranging at least some of the meetings.

What possible meeting schedule might cause this to happen? How many members are there in this club, at the least?