How would one modify Euclid's proof of infinite primes to generate a sequence of distinct prime numbers only? How can you prove that each element in the sequence generated by the modified algorithm is a prime number different from all the previous primes in the sequence?
Example- In this algorithm,Given the current list of primes p1 , … pn , you multiply them together then add 1 . This gives a number x=q1..qn +1. Here x need not be prime and I was wondering how to modify the algorithm so the "x's" generated as such would be prime numbers
Suppose you already have a list of primes $\{p_1,p_2,...p_n\}$, then $s=p_1p_2...p_n+1$does not have $p_1,...,p_n$ as prime divisors. So any prime divisor,call it $p_{n+1}$ of $s$ is a new primes not in the list of primes above. So then you can make a new list $\{p_1,...,p_n,p_{n+1}\}$ and do the same thing with the new list. Note that product of all prime numbers up to 1001 is greater than the number of atoms in the universe. It would take a computer so long to deduce if that number plus 1 is a prime or not that the universe will have ended.