\begin{definition}[Definition:Legendre's Constant]
'''Legendre's constant''' (or the '''Legendre constant''') is a mathematical constant conjectured by {{AuthorRef|Adrien-Marie Legendre}} to specify the prime-counting function $\map \pi n$.
Legendre conjectured in $1796$ that $\map \pi n$ satisfies:
:$\ds \lim_{n \mathop \to \infty} \map \pi n - \frac n {\map \ln n} = B$
where $B$ is '''Legendre's constant'''.
If such a number $B$ exists, then this implies the Prime Number Theorem.
{{AuthorRef|Adrien-Marie Legendre|Legendre}}'s guess for $B$ was about $1 \cdotp 08366$.
{{OEIS|A228211}}
Later, {{AuthorRef|Carl Friedrich Gauss}} looked at this problem and thought that $B$ might actually be lower.
In $1896$, {{AuthorRef|Jacques Salomon Hadamard}} and {{AuthorRef|Charles de la Vallée Poussin}} independently proved the Prime Number Theorem and showed that $B$ is in fact equal to $1$.
{{AuthorRef|Adrien-Marie Legendre|Legendre}}'s first guess of $1 \cdotp 08366 \ldots$ is still (incorrectly) referred to as '''Legendre's constant''', even though its "correct" value is in fact exactly $1$.
Hence it is only now of historical importance.
{{NamedforDef|Adrien-Marie Legendre|cat = Legendre}}
\end{definition}