I consider the following ascending sequence of prime numbers

2, 3, 5, 7, 11, 13, 17, 19

All the prime numbers from 0 to 19 are 2, 3, 5, 7, 11, 13, 17, 19. The question I handle now, is, exists there a prime number greater than 19? It is well know that such a prime number exists, e.g. 23 or 29 or 31 ... But I assume, that this knowledge is not available and I try to prove, that a prime number greater than 19 exists. The proof is written in a sense which can be generalized to an arbitrary sequence of prime numbers P1 < P2 <... < PN, with P _{1 } = 2, P_{2} = 3 ....Let's start with the sequence 2, 3, 5, 7, 11, 13, 17, 19. I am looking if there exists a prime number greater than 19. With the numbers 2, 3, 5, 7, 11, 13, 17, 19 I construct the number 2 + 3 * 5 * 7 *11 * 13 * 17 * 19 = 4849847 and I look first, if 4849847 is a prime number. The symbol * stands for multiplication. |

The following statements are true

2 + 3 = 5 is a prime number

2 + 3 * 5 = 17 is a prime number

2 + 3 * 5 *7 = 107 is a prime number

The idea is as follows.

The number 2 + 3 * 5 * 7 *11 * 13 * 17 * 19 can not be divided by 2, 3, 5, 7, 11, 13, 17, 19.

Therefore also 4849847 can not be divided by 2, 3, 5, 7, 11, 13, 17, 19.

The numbers 2, 3, 5, 7, 11, 13, 17, 19

form a sequence of ascending prime numbers, there are no other prime numbers between 2, 3, 5, 7, 11, 13, 17, 19.

The number 4849847 can not be divided by 2, 3, 5, 7, 11, 13, 17, 19 and if it is not a prime number,

then there exists a decomposition of the form 4849847 = X * Y with numbers X and Y, which are smaller than 4849847

The numbers X and Y can not be divided by 2, 3, 5, 7, 11, 13, 17, 19, otherwise the product X * Y = 4849847 is divisible by at least one of the numbers 2, 3, 5, 7, 11, 13, 17, 19, which has been excluded.

If X and Y are not prime numbers, then we can split again X and Y into factors of numbers and the preceding reasoning can be applied to the new factors.

The procedure must terminate, because the starting number 4849847 is finite. Termination means, at one point there exist a number X, which must be a prime number, and 4849847 is divisible by X.

If X is a prime number greater than 19, than we found a prime number for which we are looking for.

If X is a prime number less or equal to 19, then X is one of the numbers 2,3,5,7,11,13,17, 19. But because 4849847 is divisible by X than it must be divisible by one of the numbers 2,3,5,7,11,13,17,19, which has been excluded.

The above mentioned decomposion of + 3 * 5 * 7 *11 * 13 * 17 * 19 = 4849847 can be generalized to an arbitrary number P

Let PN be an arbitrary large prime number. I arrange all existing prime numbers less than PN in the way: P1 < P2 <... < PN.

It is P1 = 2, P2 = 3...

The sum of P1 + (P2 * ... * PN) is a number that is not divisible by the primes P1 to PN.

According to the above reasoning, we can conclude that P1 + (P2 * ... * PN) is itself a prime number or that there is a product decomposition of the form P1 + (P2 * ... * PN) = X1 * ... * XK that contains at least a prime number XI, which must be greater than PN.

Some samples:

2 + (3*5*7*11*13*17*19*23*29*31*37*41) = 152125131763607, is it a prime number? the above reasoning shows, there must exist a prime P which is greater than 41.

2+(3*5*7*11*13*17*19*23*29*31*37*41*43*47*53*59*61*67*71) = 278970415063349480483707697, is it a prime number? The above reasoning shows, there must exist a prime number P which is greater than 71.