Definition:Chebyshev-Sylvester Constant

Definition

The Chebyshev-Sylvester constant is a mathematical constant $\alpha$ which was shown by Pafnuty Lvovich Chebyshev and James Joseph Sylvester that for sufficiently large $x$, there exists at least one prime number $p$ satisfying:

$x < p < \paren {1 + \alpha} x$

The number $\alpha$ was demonstrated to be $0 \cdotp 092 \ldots$

Source of Name

This entry was named for Pafnuty Lvovich Chebyshev and James Joseph Sylvester.

Historical Note

In $1896$, Jacques Salomon Hadamard and Charles de la Vallée Poussin independently proved the Prime Number Theorem.

This demonstrated that the inequality defining this constant is true for all $\alpha > 0$ for sufficiently large $x$,.

Hence the Chebyshev-Sylvester constant is now only of historical interest.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,092 \ldots$

• Weisstein, Eric W. "Chebyshev-Sylvester Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chebyshev-SylvesterConstant.html