Euclid's theorem states:
Consider any finite list of prime numbers p1, p2, ..., 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 = p1p2...pn. Let q = P + 1. Then q is either prime or not:
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.
Consider any finite list of prime numbers p1, p2, ..., 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 = p1p2...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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