According to the fundamental theorem of arithmetic (unique factorization theorem), you can write every number as the product of some prime numbers, for example $33 = 11 \cdot 3$.
However, how can you do this when you're dealing with a prime number? If you write $29 = 29 \cdot 1$ you use 1 and that isn't a prime number. Should you just write $29 = 29^1$?
A single number, like $31$ or $7$, is in fact a product as far as mathematics is concerned. It is a product of $1$ integer.
Indeed, you can even have a product of $0$ integers. This is defined to be $1$, because $1$ is the identity element of multiplication. (See Qiaochu's comment.)
When we say that an integer has a unique prime factorization, we mean it can be written as a product of some nonnegative integer number of primes. Thus, "$2 \cdot 2 \cdot 23$", "$31$", "$7$", and "$\quad$" are all valid prime factorizations.