What is the largest number of distinct positive integers you can have such that most of their pairwise differences are prime?
For example, among (2, 4, 6, 11, 13, 15) there are 15 pairwise differences, of which 10 are prime.
Steve is in charge of designing a wall-hanging calendar. Each month is allocated a grid of 5 X 7 squares, labeled Sunday thru Saturday across the top. The problem is, Steve hates to put two dates in the same square on the calendar, necessary when the month spans parts of six weeks. Is it possible for Steve to find a year when he never has to put two dates in the same square? What is the most double-date squares he would ever need for a single year?