is the statment valid?

Question

it follow the theorem  that therer are infinit prime number pi from i=1 to N .Pj+1 .is a prime for every positive integer N?explaine

Answer 1

There are an infinite number of primes when N approaches infinity. Adding 1 to all but at most two prime numbers produces an even number, which, by definition, is composite because 2 is a divisor. The only prime number exceptions are 1 and 2 which produce another prime number when increased by 1. So P sub j is only prime when P sub j = 2. So the statement that (P sub j) + 1 is  a prime for every positive integer j in N is not valid. If N=10, then P1=1, P2=2, P3=3, P4=5, P5=7 (maximum i is 5). (It is debatable whether 1 is considered to be a prime P1.) So, the statement is false for j>2.