Bertrand's postulate facts for kids

Bertrand's postulate is a cool idea in mathematics that says something special about prime numbers. It states that if you pick any whole number n that is bigger than 3, you will always find at least one prime number p that is between n and 2n − 2.

For example, if n is 4, then 2n − 2 is 6. The postulate says there's a prime between 4 and 6. That prime is 5! If n is 10, then 2n − 2 is 18. The postulate says there's a prime between 10 and 18. Examples are 11, 13, or 17.

What is Bertrand's Postulate?

This idea, or "postulate," is about how prime numbers are spread out. Prime numbers are special because they can only be divided evenly by 1 and themselves (like 2, 3, 5, 7, 11, etc.).

The postulate basically guarantees that there's always a new prime number waiting for you not too far from any number you pick (as long as it's bigger than 3).

Who First Thought of This Idea?

The statement was first made in 1845 by a French mathematician named Joseph Bertrand. He didn't prove it completely, but he checked his idea for all numbers up to 3 million (3,000,000). That's a lot of checking!

How Was It Proven True?

Even though Bertrand checked many numbers, checking isn't the same as proving. A proof shows it's true for ALL numbers, no matter how big.

• Pafnuty Chebyshev: The postulate was fully proven to be true in 1850 by a Russian mathematician named Pafnuty Chebyshev. Because he proved it, the postulate is also known as the Bertrand-Chebyshev theorem or Chebyshev's theorem.
• Srinivasa Ramanujan: Later, a brilliant Indian mathematician named Srinivasa Ramanujan found a simpler way to prove it. He even used this proof when he discovered his own special set of primes called Ramanujan primes.
• Paul Erdős: In 1932, a Hungarian mathematician named Paul Erdős published an even simpler proof. Mathematicians are always looking for easier and more elegant ways to show something is true!

Why Is This Postulate Important?

Bertrand's Postulate is important in number theory, which is the study of numbers and their properties. It helps mathematicians understand how prime numbers are distributed among all the other numbers. It's a fundamental result that has been used in other areas of mathematics.

See also

In Spanish: Postulado de Bertrand para niños