The Prime Number Theorem was conjectured by Gauss from looking (very hard, one can presume...) at a table of the primes $\leq10^6$. It is not with too much effort that one can read his Disquisitioned as a set of tricks to determine primality with as little work as possible, and one can understand the motivation: he was his own computer, in a way :P

(I don't know where Legendre got the statement from, but he must surely have had tables of primes too!)

The Prime Number Theorem was conjectured by Gauss from looking (very hard, one can presume...) at a table of the primes $\leq10^6$. It is not with too much effort that one can read his Disquisitiones as a set of tricks to determine primality with as little work as possible, and one can understand the motivation: he was his own computer, in a way :P

(I don't know where Legendre got the statement from, but he must surely have had tables of primes too!)