Thread: Find the Problem with this Proof

My task is to find the flaw with the following given proof:

This "proof" attempts to prove that there do not exist any primes that are greater than 101.

Proof:

Let n > 101. Then it is evident if n is even, it's definitely not a prime.

If n is odd, then (n + 1)/2 and (n - 1)/2 are integers. Let x = (n + 1)/2 and y = (n - 1)/2. Then, n = x^2 - y^2 = (x + y)*(x-y), thus odd n is not a prime.

And thus the proof is complete proving that there do not exist any primes > 101.

Q.E.D?

Originally Posted by Ideasman

My task is to find the flaw with the following given proof:

This "proof" attempts to prove that there do not exist any primes that are greater than 101.

Proof:

Let n > 101. Then it is evident if n is even, it's definitely not a prime.

If n is odd, then (n + 1)/2 and (n - 1)/2 are integers. Let x = (n + 1)/2 and y = (n - 1)/2. Then, n = x^2 - y^2 = (x + y)*(x-y), thus odd n is not a prime.

And thus the proof is complete proving that there do not exist any primes > 101.

Q.E.D?

For this to work would require that (x+y) and (x-y) are both integers !=1,
but (x+y)=n and (x-y)=1, so n=(x+y)(x-y) is a factorosation, but not
into factors different from 1 and n.

RonL

Originally Posted by Ideasman

My task is to find the flaw with the following given proof:

This "proof" attempts to prove that there do not exist any primes that are greater than 101.

Proof:

Let n > 101. Then it is evident if n is even, it's definitely not a prime.

If n is odd, then (n + 1)/2 and (n - 1)/2 are integers. Let x = (n + 1)/2 and y = (n - 1)/2. Then, n = x^2 - y^2 = (x + y)*(x-y), thus odd n is not a prime.

And thus the proof is complete proving that there do not exist any primes > 101.

Q.E.D?

The definition of a prime is that is has no non-trivial proper divisors. You need to show that (x+y) is proper and (x-y) is not trivial to complete the proof. But that last step is wrong.