Euclid's theorem states:

Consider any finite list of prime numbers

Does this theorem also hold if you let

Wouldn't P-1 also be necessarily a new prime number? And if so, it and P+1 would be a set of twin primes.

So the proof would be:

Assume there are a finite number of twin primes such that P(n+1) - P(n) = 2

Then, from the final set of twin primes, choose the larger of these two primes P(n+1). Calculate S=p1p2...P(n+1). So you now have a product of all primes up to P(n+1). Call this S. S + 1 is a prime number and so is S - 1. This is a new set of twin primes not in our original list, thus there cannot be a finite list of twin primes.

Of course if S - 1 is not prime, then this falls apart.

Consider any finite list of prime numbers

*p*1,*p*2, ...,*pn*. It will be shown that at least one additional prime number not in this list exists. Let*P*be the product of all the prime numbers in the list:*P*=*p*1*p*2...*pn*. Let*q*=*P*+ 1. Then*q*is either prime or not:- If
*q*is prime, then there is at least one more prime that is not in the list. - If
*q*is not prime, then some prime factor*p*divides*q*. If this factor*p*were in our list, then it would divide*P*(since*P*is the product of every number in the list); but*p*divides*P*+ 1 =*q*. If*p*divides*P*and*q,*then*p*would have to divide the difference[3] of the two numbers, which is (*P*+ 1) −*P*or just 1. Since no prime number divides 1,*p*cannot be on the list. This means that at least one more prime number exists beyond those in the list.

**My question:**Does this theorem also hold if you let

*q*=*P*- 1?Wouldn't P-1 also be necessarily a new prime number? And if so, it and P+1 would be a set of twin primes.

So the proof would be:

Assume there are a finite number of twin primes such that P(n+1) - P(n) = 2

Then, from the final set of twin primes, choose the larger of these two primes P(n+1). Calculate S=p1p2...P(n+1). So you now have a product of all primes up to P(n+1). Call this S. S + 1 is a prime number and so is S - 1. This is a new set of twin primes not in our original list, thus there cannot be a finite list of twin primes.

Of course if S - 1 is not prime, then this falls apart.

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