Is the sum of the coefficients of the polynomial interpolation of the data $(1,p_1),(2,p_2),...,(n,p_n)$ for some positive integer $n$ (where $p_n$ is the $n$th prime) always equal to two? I've verified up to $n=6$.
$n = 2: y = x + 1, \text{sum} = 1 + 1 = 2$
$n = 3: y = \frac{1}{2}x^2-\frac{1}{2}x+2, \text{sum}=2$ . . .
Yes. The sum of the coefficients of a polynomial is just the polynomial evaluated at 1. So since the value of the interpolation at 1 is $p_1=2$, the sum of the coefficients will always be 2.
To see this, just consider:
If $$p(x)=\sum_{i=0}^n a_ix^i,$$ then $$p(1)=\sum_{i=0}^n a_i(1)^i=\sum_{i=0}^n a_i.$$