Richard Heathfield said:
... All the mathematics teachers I ever had believed 1 to
be prime. ...
People who quible over whether 1 is prime or not are
usually people for whom the difference is entirely
superficial! ;-)
The common definition of prime in schools is actually the
definition attributed to a related, but distinct, property of
'irreducability'.
In the domain of Gaussian integers, 5 is irreducable in
that it has no factors other than itself and unity
(ignoring sign), yet it is not prime!
Depending on which mathematician you talk to, a prime
is often defined as an element that if it divides a
product of 2 elements, then it also divides at least
one of the elements in the product. Thus, 6 is not a
prime. Not because it has the factors of 2 and 3, but
because it divides 12 without dividing 3 or 4.
Fact is, units fit nicely into either definition. What
matters is when you start to apply the definition to
build theorems. Sometimes units are incidental, mostly
they're a hinderance. Hence it is more common to
explicitly exclude them.
[Mathematicians don't care which definition you use.
What matters is whether you use the definition
rigorously.]
My point is that dumbing things down is a common and
valuable teaching tool. The concept of primes is a
good example. It is the irreducable property of primes
that sparks people's interest. Their real meaning
is unwieldy and... boring!