Bertrand-Chebyshev Theorem/Historical Note

Historical Note on Bertrand-Chebyshev Theorem

The Bertrand-Chebyshev Theorem was first postulated by Bertrand in $1845$. He verified it for $n < 3 \, 000 \, 000$.

It became known as Bertrand's Postulate.

The first proof was given by Chebyshev in $1850$ as a by-product of his work attempting to prove the Prime Number Theorem.

After this point, it no longer being a postulate, Bertrand's Postulate was referred to as the Bertrand-Chebyshev Theorem.


In $1919$, Srinivasa Ramanujan gave a simpler proof based on the Gamma function.


In $1932$, Paul Erdős gave an even simpler proof based on basic properties of binomial coefficients. That proof is the one which is presented here.


Sources

  • 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.31$: Chebyshev ($\text {1821}$ – $\text {1894}$)
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bertrand's postulate
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bertrand's postulate
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bertrand's postulate
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