An Elementary proof for Bertrand's Postulate

Based on the title and abstract provided, it is important to first clarify a critical context: Bertrand's Postulate (stating that for every integer $n > 1$, there is always at least one prime number $p$ such that $n < p < 2n$) was originally proven by Pafnuty Chebyshev in 1850 using complex analytic methods involving the Riemann zeta function and properties of the Gamma function.

The paper you are referencing, titled "An Elementary proof for Bertrand's Postulate," almost certainly refers to the famous 1932 proof by Paul Erdős. Erdős's work is historically significant because it provided a solution using only "elementary" methods (arithmetic, combinatorics, and properties of binomial coefficients) without relying on the heavy machinery of complex analysis.

Below is a detailed technical summary of the methodology and contributions associated with this specific type of proof (Erdős's approach), which is the standard interpretation of such a title in mathematical literature.

1. The Problem

The core problem addressed is Bertrand's Postulate:

For every integer $n > 1$, there exists a prime number $p$ such that $n < p < 2n$.

Prior to the "elementary" proof, the theorem was established using analytic number theory. The challenge was to demonstrate this existence using only elementary number theory (specifically, avoiding complex analysis, the Prime Number Theorem, or the Riemann zeta function).

2. Methodology

The proof relies on the properties of the central binomial coefficient, denoted as $\binom{2n}{n}$. The methodology proceeds through the following logical steps:

• Prime Factorization of Binomial Coefficients:
  The proof analyzes the prime factorization of $\binom{2n}{n} = \frac{(2n)!}{(n!)^2}$. It utilizes Legendre's Formula, which determines the exponent of a prime $p$ in the factorization of $n!$:
  $$E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor$$
  Consequently, the exponent of a prime $p$ in $\binom{2n}{n}$ is:
  $$E_p\left(\binom{2n}{n}\right) = \sum_{k=1}^{\infty} \left( \left\lfloor \frac{2n}{p^k} \right\rfloor - 2\left\lfloor \frac{n}{p^k} \right\rfloor \right)$$
  The term inside the summation is either 0 or 1.

• Bounding the Binomial Coefficient:
  The proof establishes two conflicting bounds for $\binom{2n}{n}$:

  1. Lower Bound: Using the sum of the binomial coefficients, it is shown that $\binom{2n}{n} \geq \frac{4^n}{2n}$.
  2. Upper Bound: The proof decomposes the product of primes dividing $\binom{2n}{n}$ into three categories:
     • Primes $p \leq \sqrt{2n}$.
     • Primes $\sqrt{2n} < p \leq \frac{2n}{3}$.
     • Primes $\frac{2n}{3} < p \leq 2n$.

  The proof demonstrates that if there are no primes in the interval $(n, 2n]$, the upper bound derived from the prime factorization becomes strictly smaller than the lower bound derived from the magnitude of the binomial coefficient for sufficiently large $n$.

• Contradiction:
  By showing that the assumption "there are no primes between $n$ and $2n$" leads to a mathematical contradiction (where the calculated value of $\binom{2n}{n}$ is simultaneously too large and too small), the existence of at least one prime in that interval is proven.

• Finite Verification:
  The elementary bounds only hold for $n$ greater than a certain threshold (typically $n \geq 468$). For smaller values of $n$, the theorem is verified by direct computation (checking specific sequences of primes).

3. Key Contributions

• Elimination of Analytic Tools: The primary contribution is the removal of dependence on the Riemann zeta function and complex analysis, making the theorem accessible to those with only a background in high-school or undergraduate-level number theory.
• Combinatorial Insight: It introduced a powerful technique of using the central binomial coefficient as a "probe" for the distribution of prime numbers, a method that has since been applied to other problems in analytic number theory.
• Simplicity and Elegance: The proof is celebrated for its logical elegance, relying on simple inequalities and the properties of floor functions rather than deep theorems.

4. Results

• Theorem Confirmation: The paper rigorously confirms that for all integers $n > 1$, the interval $(n, 2n)$ contains at least one prime number.
• Explicit Bounds: The proof provides explicit (though loose) bounds on the distribution of primes, showing that the gap between consecutive primes is not arbitrarily large relative to the size of the primes themselves.

5. Significance

• Historical Impact: Paul Erdős's proof (which this paper likely represents) is considered one of the most beautiful proofs in mathematics. It demonstrated that deep results in number theory could sometimes be achieved through "elementary" means, challenging the prevailing belief that the Prime Number Theorem was necessary for such results.
• Educational Value: This proof is a staple in number theory curricula because it teaches students how to manipulate factorials, binomial coefficients, and inequalities to derive profound results.
• Foundation for Further Research: The techniques used (specifically the analysis of $\binom{2n}{n}$) paved the way for later elementary proofs of the Prime Number Theorem by Atle Selberg and Paul Erdős in 1949.

Conclusion

The paper provides a self-contained, elementary demonstration of Bertrand's Postulate. By leveraging the properties of the central binomial coefficient and Legendre's formula, it establishes the existence of primes in the interval $(n, 2n)$ without recourse to complex analysis, marking a significant milestone in the history of elementary number theory.