These are from lecture and I don't understand them thoroughly. Any tips that could help me to understand this better would be greatly appreciated. Here are the proofs.

1) A positive integer is called prime if its only positive divisors are 1 and itself. Otherwise it is called composite.

2) There are infinitely many primes.

These are from lecture and I don't understand them thoroughly. Any tips that could help me to understand this better would be greatly appreciated. Here are the proofs.

1) A positive integer is called prime if its only positive divisors are 1 and itself. Otherwise it is called composite.

2) There are infinitely many primes.
What don't you understand; the definition of a prime or that there are
infinitly many of them, or the proof of the latter.

RonL

3. ## proofs of these statements

I got some messy notes from a classmate on these and I can't figure out the proofs of the statements. That was all that I wanted to see. Prove both statements. Thanks

I got some messy notes from a classmate on these and I can't figure out the proofs of the statements. That was all that I wanted to see. Prove both statements. Thanks
The first statement is just a definition, there's nothing to prove.

For the second, the most famous (and perhaps first) proof for the infinitude of primes was given by Euclid. It may be the proof you want.

See here or here

at the first link, other proofs besides Euclid's are shown.

1) A positive integer is called prime if its only positive divisors are 1 and itself. Otherwise it is called composite.
This is a side bar.
I join many others who disagree with #1 as a definition of prime. I know that many use it.
I even heard Keith Devlin it use on National Public Radio. However, that definition makes 1 a prime.

6. What do you think is the best definition of a prime?

7. Originally Posted by DivideBy0
What do you think is the best definition of a prime?
A positive integer with exactly two divisors is prime.

8. Originally Posted by Plato
A positive integer with exactly two divisors is prime.
as far as i remember, the additional condition for a number to be prime is that it should be greater than 1. that what makes 1 not a prime.. Ü