# Bertrand-Chebyshev Theorem/Historical Note

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## 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 ($\text {1821}$ – $\text {1894}$)