Bertrand's Postulate — Definition, Formula & Examples

Bertrand's Postulate is a proven theorem stating that for every positive integer

n

, there is always at least one prime number between

n

and

2n

(inclusive of

2n

For every integer $n \geq 1$ , there exists a prime $p$ such that $n < p \leq 2n$ . Conjectured by Joseph Bertrand in 1845 and first proved by Pafnuty Chebyshev in 1852, this result is a theorem despite retaining its historical name as a "postulate."

Key Formula

\forall\, n \in \mathbb{Z}^{+},\; \exists\, p \text{ prime such that } n < p \leq 2n

Where:

$n$ = Any positive integer (n ≥ 1)
$p$ = A prime number guaranteed to exist in the interval (n, 2n]

How It Works

The theorem guarantees that primes cannot be too sparse: the gap between consecutive primes never grows so large that you can double a number without passing through a prime. To apply it, pick any positive integer

n

; you are assured a prime exists in the half-open interval

(n, 2n]

. Stronger results, such as those by Ramanujan and Erdős, later gave simpler proofs and tighter bounds. Erdős's 1932 proof, published when he was 19, is celebrated for its elegance using properties of binomial coefficients.

Worked Example

Problem: Verify Bertrand's Postulate for n = 10 by finding a prime p with 10 < p ≤ 20.

Identify the interval: We need a prime strictly greater than 10 and at most 20.

10 < p \leq 20

List integers in the range: The integers to check are 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

Find primes: Among these, 11, 13, 17, and 19 are all prime. Any one of them satisfies the postulate.

Answer: The primes 11, 13, 17, and 19 all lie in the interval (10, 20], confirming Bertrand's Postulate for n = 10.

Why It Matters

Bertrand's Postulate is a foundational result in analytic and elementary number theory. It underpins arguments about prime gaps and is frequently used as a lemma in combinatorics problems—for instance, proving that the central binomial coefficient

\binom{2n}{n}

has a prime factor in a certain range. It also appears in competition mathematics and serves as a gateway to deeper results like the Prime Number Theorem.

Common Mistakes

Mistake: Assuming "postulate" means the result is unproven or taken as an axiom.

Correction: Bertrand's Postulate is a fully proven theorem. The name is historical—Bertrand conjectured it, and Chebyshev proved it rigorously in 1852.