Degree of polynomial interpolating the primes The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = \frac{x^2}{2}-\frac{x}{2}+2 \;.
$$
Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For example:
$$
p_5(x) = \frac{x^4}{8}-\frac{17
   x^3}{12}+\frac{47
   x^2}{8}-\frac{103 x}{12}+6 \;.
$$
One can check that 
\begin{eqnarray}
p_5(1) &=& 2 \\
p_5(2) &=& 3 \\
p_5(3) &=& 5 \\
p_5(4) &=& 7 \\
p_5(5) &=& 11 \;.
\end{eqnarray}
My question is:

Q. Is the degree of $p_n(x)$ ever strictly less than $n{-}1$, for any $n$?

The answer to Q is positive if a "coincidence" occurs,
such that a smaller degree
polynomial captures those $n$ prime points.
Do such coincidences ever occur?

Not really an answer, but consider a more general question:

Does the polynomial interpolating $n$ consecutive primes $p_{m+1},\dots,p_{m+n}$ always have maximum degree $n-1$?

The answer is a strong no because $3,5,7$ and $251,257,263,269$ are consecutive primes in arithmetic progression.
Small examples are known for $3 \le n \le 6$. See Wikipedia.
The polynomial interpolating the four consecutive primes $17, 19, 23, 29$ has degree $2$, not $3$. So does the polynomial interpolating the four consecutive primes $p_{m+1},\dots,p_{m+4}$ for $m \in \{6,10,12,17,21,48,57,68,69,74,84,90,103,110,115,121,122,126,131,172,181\}$.
