Wilson prime facts for kids

A Wilson prime is a special type of prime number. Imagine a prime number, which is a whole number greater than 1 that can only be divided evenly by 1 and itself (like 2, 3, 5, 7, etc.). A prime number p is called a Wilson prime if a very specific math rule works for it.

This rule involves something called a factorial. A factorial of a number (like 4!) means multiplying all the whole numbers from 1 up to that number. So, 4! is 1 × 2 × 3 × 4 = 24.

For a prime number p to be a Wilson prime, if you take the number just before p (which is p-1) and find its factorial, then add 1 to that result, the new number must be perfectly divisible by p multiplied by itself (which is p squared, or p²).

In math terms, it looks like this: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\left(p-1\right)! + 1}{p^2} If this calculation gives you a whole number (an integer) without any remainder, then p is a Wilson prime!

The idea of Wilson primes was first described by a mathematician named Emma Lehmer.

What is a Prime Number?

A prime number is a whole number greater than 1 that has only two positive divisors: 1 and itself. Think of it like a building block for all other numbers.

• For example, 7 is a prime number because you can only divide it evenly by 1 and 7.
• But 6 is not a prime number because you can divide it by 1, 2, 3, and 6.

Prime numbers are very important in mathematics and are used in many areas, including computer science and cryptography (making codes).

Understanding Factorials

The exclamation mark "!" in math means "factorial." When you see a number followed by "!", it means you multiply that number by every whole number smaller than it, all the way down to 1.

• For example, 3! (read as "3 factorial") is 3 × 2 × 1 = 6.
• 5! (read as "5 factorial") is 5 × 4 × 3 × 2 × 1 = 120.

So, for a Wilson prime, we look at the factorial of the number just before our prime number. If our prime number is 5, we look at (5-1)! which is 4!. And 4! = 24.

The Wilson Prime Rule Explained

Let's use an example to see how the rule works for a Wilson prime. The number 5 is a Wilson prime. 1. Our prime number p is 5. 2. We need to find (p-1)!, which is (5-1)! = 4!. 3. 4! = 4 × 3 × 2 × 1 = 24. 4. Now, we add 1 to this result: 24 + 1 = 25. 5. Next, we need to divide this by p squared, which is 5² = 25. 6. So, we calculate 25 ÷ 25 = 1. Since the answer is a whole number (1), 5 is indeed a Wilson prime!

Known Wilson Primes

Wilson primes are very rare! So far, only three Wilson primes have been found:

• 5
• 13
• 563

Scientists have checked numbers much larger than these. If there are any other Wilson primes, they must be incredibly big, larger than 500,000,000! This shows how special and uncommon these numbers are.

The Wilson Prime Conjecture

A conjecture in mathematics is like a very strong guess or an idea that mathematicians believe is true, but they haven't been able to prove it yet. For Wilson primes, mathematicians have a conjecture that there are an infinite number of them. This means they believe there's no end to how many Wilson primes exist, even if we've only found a few so far.

They also have ideas about how many Wilson primes might be found within a certain range of numbers, but this is still a topic of research.

Connection to Wilson's Theorem

Wilson primes are closely related to a more general rule in number theory called Wilson's theorem. Wilson's theorem states that for any prime number p, the number (p-1)! + 1 will always be perfectly divisible by p.

• For example, if p is 7:

* (7-1)! + 1 = 6! + 1 = (720) + 1 = 721. * 721 divided by 7 is 103, with no remainder. So, Wilson's theorem holds true for 7.

Wilson primes are special because they take this a step further. Not only is (p-1)! + 1 divisible by p, but for Wilson primes, it's also divisible by p squared (p²). So, Wilson primes are like "super" primes when it comes to Wilson's theorem!

Related Math Topics

• Wieferich prime
• Wall-Sun-Sun prime
• Wolstenholme prime

See also

In Spanish: Número primo de Wilson para niños