Find the first 20 primes found by the classical proof of the infinitude of the set of primes

Begin with P={2}; then form,m, the sum of 1 with the product overall elements of P. Place the smallest prime factor of m into P and repeat.

Suppose p = {p1,p2,…,pr}, then m = 1+ p1p2p3…pr.

Example:

2 is prime and 2+1 = 3 is prime;

2 * 3 +1 = 7 is prime;

2 * 3 * 7 +1 = 43 is prime;

2 * 3 * 7 * 43 +1 = 1807 = 13 * 139, then 13 is the prime;

Thus the first 5 prime number found by the classical proof is {2,3,7,43,13}.

So how to use this proof to find the first 20 prime in Mathematica?
Thank you.

Clear["Global`*"]

n = 20;

(p = Nest[Join[#, {FactorInteger[1 + Times @@ #][[1, 1]]}] &,
    {2}, n - 1]) // AbsoluteTiming

(* {57.663, {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 
  11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957}} *)

Length[p]

(* 20 *)

And @@ (PrimeQ /@ p)

(* True *)