Carmichael number facts for kids

A Carmichael number is a special kind of composite number. A composite number is a whole number that can be divided evenly by numbers other than 1 and itself. Carmichael numbers are unique because they act a bit like prime numbers in a certain math test.

Prime numbers are whole numbers greater than 1 that can only be divided evenly by 1 and themselves (like 2, 3, 5, 7). There's a famous rule for prime numbers called Fermat's Little Theorem. It says that for any prime number p, if you pick another number b that doesn't share any common divisors with p (except 1), then b raised to the power of (p-1) will have a special relationship with p.

Usually, if a number is composite, it won't pass this test. But Carmichael numbers are tricky! They are composite, yet they pass this test, just like prime numbers do. This makes them quite rare and interesting in the world of number theory.

What Makes Carmichael Numbers Special?

Carmichael numbers are composite numbers that behave like prime numbers in a specific way. This behavior is related to a rule discovered by the famous mathematician Pierre de Fermat. His rule, called Fermat's Little Theorem, works for all prime numbers.

For example, if you take a prime number like 5, and another number like 3 (which doesn't share common divisors with 5), then 3 raised to the power of (5-1), which is 3 to the power of 4 (81), will leave a remainder of 1 when divided by 5.

Most composite numbers do not follow this rule. If you try it with a composite number like 4, it usually won't work. But Carmichael numbers are different. Even though they are composite, they always follow this rule, just like prime numbers. This is why they are sometimes called "pseudoprimes" – they pretend to be prime!

Who Discovered Carmichael Numbers?

These special numbers are named after a mathematician named Robert Daniel Carmichael. He was an American mathematician who studied these numbers in the early 1900s. He published his findings in 1910.

However, the first few Carmichael numbers were actually found earlier by another mathematician. Václav Šimerka, a Czech mathematician, listed the first seven Carmichael numbers in 1885. He found them while working on similar mathematical problems.

The smallest Carmichael number is 561. The next ones are 1105, 1729, 2465, 2821, and 6601. These numbers are quite rare. Mathematicians have proven that there are actually an infinite number of Carmichael numbers, but they become much harder to find as they get larger.

Images for kids

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  Václav Šimerka listed the first seven Carmichael numbers

See also

In Spanish: Número de Carmichael para niños