The smallest prime number, according to primality as defined by modern mathematicians, is 2. To be prime, a number must be divisible only by itself and 1. All other numbers are known as composites.
The only positive integer lower than 2 is 1, and 1 was indeed treated as a prime number throughout the 19th century. This convention was eventually dropped as the definition of a prime number was changed to exclude 1. The reason for this change relates to the way prime numbers are used to compute positive integers. Euclid proposed that each positive integer can be expressed as the product of unique prime numbers. Treating 1, which is the multiplicative identity, as a prime number falsifies this conjecture.