What would change in mathematics if the definition of a prime number included 1?

Answer 1

There would no longer be a unique prime factorization for positive integers - for example, 12 could be expressed as both 2*2*3, 1*2*2*3, or 1*1*1*1*2*2*3, or any other number. A lot of mathematical proofs that depended on this property, or a number of other properties of prime numbers, would have to be rewritten. Inevitably, someone would come up with some other definition, such as "prime rib numbers", which would be defined as "prime numbers not including 1", and then people would start using these "prime rib numbers" instead. The definitions in mathematics are anything but arbitrary - all of them have been introduced for a reason. It's possible to make up your own definitions, but unless there were a good reason to use them, no one would accept them - the fact that every mathematician knows what a Prime number is means that the definition of prime numbers is a good one.