About Euclid's proof of infinite primes..... I was checking that if product of first n primes+1 gives a prime again is true to how many n
For example $$2+1=3$$ is a prime$$2\times 3+1=7$$ is a prime$$2\times 3\times 5+1=31$$ is a prime$$2\times 3\times 5\times 7+1=211 $$ is a prime$$2\times 3\times 5\times 7\times 11+1=2311$$ is a prime $$2\times 3\times 5\times 7\times 11\times 13+1=30031$$ is composite
So prime chain is broken and further steps will give composite no.s only
Now as I understood from proof of infinite primes Euclid said multiply all primes and add 1 and you will get another prime. It proves that there are infinite primes
Then my question is how can we guarantee that the resulting number is prime??
Or it should be that we will get another prime dividing the resulting number
Please help me to clear my confusion!!!!

Neither Euler nor Euclid said any such thing. Euclid wrote that if $p_1,..,p_n$ are primes and $q$ is $1$ more than their product , then any prime divisor of $q$ is  a prime not equal to any of $p_1,...,p_n.$
If   $p$ is the least $m>1$ that divides $q$ then $p$ is a prime divisor of $k.$ So $q$ does indeed have a prime divisor.
