Prove that there are infinitely many primes

timer Asked: Aug 14th, 2014
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Question Description

How can I prove that there are infinitely many primes? Thanks for the help!

Tutor Answer

Solution E
School: Boston College

To prove that there are infinitely many prime numbers we will be using Euclid's proof to do this;

normally it is known that a prime number is any number that is  natural and has only/ exactly two divisors that is itself and 1.


taking an assumption trhat there are infinitely many prime numbers that is x1,x2,x3,x4.....xn  and constructing a number X in  such a way that it is one more than the product of all finitely many primes then we will correctly conclude that;

X= x1x2......Xn+1

this would imply that the number X will have a remainder 1 when we divide by any prime number Xi where i=1,.....n

which would make it prime number just as long as X≠ 1

we know that 2 is a prime implying that Xi is non empty. This  implies that X is greater than 1 and would therefore have two distinct divisors, (implying it is actually a prime number)

from the definition we can clearly see that X is strictly greater that any of the Xi's

 this would actually contradict our earlier assumption that there are infinitely many prime numbers

 This contradicts our assumption that there are finitely many

prime numbers. Therefore, there are infinitely many prime numbers.

Alternatively, one can leave out the assumption and let p1, . . . , n be any arbitrary

finite list of prime numbers. Then the conclusion would state that for

any finite list of prime numbers, it is possible to construct a larger prime than

any on the list. This method uses induction.

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