Infinite Number of Prime Numbers and a False Start to a Formula of Prime Number Generation

Photograph taken by Mark A. Wilson (Wilson44691, Department of Geology, The College of Wooster).[1], Public domain, via Wikimedia Commons

Euclid proved there are an infinite number of prime numbers. The summary of the proof is as follows:

•  Let the list of prime numbers be designated as p1, p2, p3, p4...pn.
•  Let P equal the product of all primes, p1 through pn, or p1 x p2 x p3 x ... x pn
•  Let Q = P + 1
•  Q is either prime or composite 
•  If Q is prime, then there exists another prime number not on the list p1 through pn
  
•  If Q is composite, it would have to be divisible by a factor, p. If p is a factor of Q, it would also divide the difference of Q - P = 1 which is not possible so p is another prime not on the prior list. 

The above proof relies on Euclid's Fundamental Theorem of Arithmetic. Other proofs are also available.

Does this method used by Euclid suggest a way to generate prime numbers? For example, multiplying primes and adding one to make a new prime number. Consider the first few primes: 2, 3, 5, 7, 11, 13, 17:

First prime only: 2 + 1 = 3 (Prime)

First 2 primes: 2 x 3 + 1 = 7 (Prime)

First 3 primes 6 x 5 + 1 = 31 (Prime)

First 4 primes 30 x 7 + 1 = 211 (Prime)

First 5 primes 210 x 11 +1 = 2311 (Prime)

First 6 primes 2310 x 13 + 1 = 30031 (Not Prime, it has the factors 59 and 509)

We have a counter-example so there is no need to continue, but the situation doesn't get better:

First 7 primes 30031 x 17 + 1 = 510511 (Not Prime, it has the factors 19, 97, 277, 1843, 5263, 26869) 

So our answer is No. We cannot successfully produce new primes by multiply a series of primes and adding 1.

update 8/6/2021 - also review Fermat's little Theorem - for a prime number p, 2 raised to the pth power is 2 more than a multiple of p.