Rusty returning student stuck on this one proof:
Prove false:
There exists an integer n such that is prime.
I know to do this, I have to prove the universal negation true.
For all integers n such that is not prime.
First is not the sum of two squares. is not a perfect square unless n = 0.
Second,
This means that, for any n, is divisible by 3 and is thus a composite number. (A composite number is simply the product of two numbers, neither of which are 1, not necessarily two primes.)
-Dan
Not true! 3 is divisible by 3 but 3 is not a composite number!
A positive integer divisible by 3 is not prime only if it’s not equal to 3 itself. It this particular problem, is not prime because 3 divides it and is not equal to 1 for any integer n.
I had to learn the hard way too. Once I wanted to prove that, while twin primes exist, you could never have “triple primes” (three consecutive odd primes). I thought I had cool proof in just one line: “Any sequence of three consecutive odd integers must contain a multiple of 3 and so that number in the sequence cannot be prime.” Then someone knocked me off my pedestal by showing me that one set of triple primes does exist: 3, 5, 7. I have learnt my lesson since then!