I thought it would be interesting to look at the special case of 1 in regards to prime numbers. Along my other research I found out that 1 was removed from the list of prime numbers in the 19th century. This made me curious to see what their reasons were. The most common definition of prime number was “a number that is divisible by 1 and itself.” 1 fits this definition, but some mathematicians were not happy with the way in which 1 is different from the other prime numbers. One reason 1 isn't considered a prime, which we talked about in class, was because if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since uniqueness would be false because any n=n·1. There would be no unique factorization of primes if 1 was considered to be a prime. Also, if 1 were a prime number, there would be infinitely many ways. We could write 12 for example, as 2*2*3, or 1*2*2*3, or 1*1*1*1*1*2*2*3. Therefore, having only one way to write a number as a product of primes is very useful when doing math. To sum up, this change from 1 being considered a prime before the 19th century, was due to the fundamental theorem of arithmetic.

I know why they did it, but it still seems odd to me that mathematicians can arbitrarily remove a number that fits the definition from the set of prime numbers just because it does not fit a theorem that is later developed.

I understand what Alli is saying in regards to uniqueness and how removing "1" from the list of primes allows us to write numbers (such as her example of "12") in a unique way. However, I also seen John's point int that is does seem strange that mathematicians can just remove a number that fits the definition from the set of prime numbers. But everything is constantly changing, and we do have to allow for this to happen as new things are discovered. I guess these mathematicians decided it was more important for all other numbers to have uniqueness as opposed to the number "1" to be prime.

along these lines I think its interesting how you can they "product" of a prime number is just itself. When I think of product its NxM and not just M.

It does seem like with number theory there should be some cut and dry rules, but apparently removing 1 for the purposes of a theorem is legal. Aren't we taught that a product involves 2 numbers? I think we need to start learning these simple number theory facts earlier in our education so when we reach college such a simple concept doesn't surprise us so much!

Yeah, it would be better to say that every positive integer n is either a prime or a product of primes.

That was the problem that I had with the FTM initially. Because then prime numbers wouldn't have a prime factorization since they wouldn't have a "product" of primes, unless 1 was counted as a prime. I think it would be interesting to see if in a couple hundred years (or maybe sooner) people redefine the FTM, or primes, or products, etc., and look back on us as if our math were primitive.

It seems to me that they could have adapted the FTM to allow 1 to be classified as prime. Maybe there will be more theorems that we haven't discussed yet that also run into conflicts with 1 being called a prime. Perhaps it was just easier to classify it as not prime than to adjust all the other theorems.