How many sets of 3 or more consecutive positive odd numbers greater than 1 exist such that all of the numbers are prime?
Let us denote the number of consecutive odd integers as n.
At the outset, we shall consider the case n=3, where the elements of the set are {a,b,c}
Since a,b and c are odd and consecutive, it follows that:
For a=3, the required set is {3,5,7}. However, if a=5, then c=9 which is composite, and therefore, a contradiction.
If a>= 7, then a mod 6 cannot be 3, since a would be composite in that situation.
Thus, a mod 6 = 1, 5. Now, a mod 6 = 1, gives b mod 6= 3, so that b is composite, which is a contradiction.
Similarly, a mod 6 = 5 will always yield composite c.
Acordingly, for n=3, the only valid set is {3,5,7}. For n>=3, let the elements be {a(1), a(2), a(3), ....., a(n)}
We had shown earlier that at least one of the elements a(1), a(2) and a(3) will always be composite whenever a(1) <=5
For a(1) = 3, we have: a(4) = 9, which is composite and therefore a contradiction.
Consequently, {3,5,7} is the only set which is fully in conformity with the provisions of the problem under referencve.
Puzzle Solution
