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How I wish I'd always taught... VAMPIRE NUMBERS.

Updated: Apr 10, 2023

We have already written about the way maths cateogrises numbers including prime numbers and vampire numbers but there are many different types of numbers which evade curricula around the world. Today, we explore some of the cousins and distant relations that make up the number system: vampire numbers, narcissistic numbers, cake numbers, happy numbers, evil numbers, pronic numbers and repunit numbers.


Vampire numbers

Vampire numbers are a special type of number that can be expressed as the product of two other numbers, where the digits of those two numbers are rearranged and put together to form the original number.



To better understand vampire numbers, let's look at an example:

1260 = 21 x 60


In this example, we see that the digits of 21 and 60 can be rearranged and multiplied to give us 1260. This makes 1260 a vampire number.


The term "vampire number" is used because they can be seen as numbers that "suck the life" out of the two numbers that make them up.


But not all numbers can be vampire numbers. For a number to be considered a vampire number, it must have an even number of digits, and the two numbers that make it up must have the same number of digits.

Let's look at a few more examples of vampire numbers:

  1. 1827 = 21 x 87

  2. 102510 = 201 x 510

  3. 104260 = 260 x 401

  4. 125460 = 204 x 615

As you can see from the examples above, vampire numbers can be quite large and have a lot of factors. In fact, some vampire numbers can have multiple pairs of factors that can be rearranged to form the original number.


One interesting thing about vampire numbers is that they are quite rare. In fact, there are only a few hundred known vampire numbers in existence. This makes them a fascinating topic for mathematicians and number enthusiasts alike.


Narcissistic numbers

Narcissistic numbers are a special type of number in which the sum of each digit is raised to the power of the number of digits in the number is equal to the number itself. For example, 153 is a narcissistic number because 1³ + 5³ + 3³ = 153.


Here are a few more examples of narcissistic numbers:

  1. 1 is a narcissistic number because 1³ = 1.

  2. 8208 is a narcissistic number because 84 + 24 + 04 + 84 = 8208.

  3. 9474 is a narcissistic number because 94 + 44 + 74 + 44 = 9474.

Cake numbers

Cake numbers are a type of mathematical concept that is easy to understand and fun to play with. They are called cake numbers because they are similar to slices of a cake. Essentially, a cake number is a number that can be divided into n parts, where each part has the same integer value.


Here are a few examples of cake numbers:

  1. 12 is a cake number because it can be divided into 2, 3, 4, or 6 equal parts (2 x 6, 3 x 4, 4 x 3, or 6 x 2).

  2. 16 is a cake number because it can be divided into 2, 4, or 8 equal parts (2 x 8, 4 x 4, or 8 x 2).

  3. 24 is a cake number because it can be divided into 2, 3, 4, 6, or 8 equal parts (2 x 12, 3 x 8, 4 x 6, 6 x 4, or 8 x 3).

As you can see, cake numbers are fun and easy to work with. They can be used to teach children about division and fractions, and they can also be used in more advanced math problems. For example, cake numbers can be used to solve problems in computer science, where it is important to divide data into equal parts to ensure efficient processing.


Happy numbers

Happy numbers are a type of number that is easy to understand and fun to work with. To determine if a number is a happy number, you take the sum of the squares of its digits, and then repeat this process with the sum of the squares of the digits until you get a result of 1. If you end up with 1, the number is a happy number.


Here are a few examples of happy numbers:

  1. 7 is a happy number because 7² = 49 and 4² + 9² = 97, and 9² + 7² = 130, and 1² + 3² + 0² = 10, and 1² + 0² = 1.

  2. 19 is a happy number because 1² + 9² = 82, and 8² + 2² = 68, and 6² + 8² = 100, and 1² + 0² + 0² = 1.

  3. 139 is a happy number because 1² + 3² + 9² = 91, and 9² + 1² = 82, and 8² + 2² = 68, and 6² + 8² = 100, and 1² + 0² + 0² = 1.

As you can see, happy numbers are fun to work with because you get to repeat the process of squaring digits until you get a result of 1. They are named happy numbers because the result of the process is always 1, which is considered a happy outcome.


Evil numbers

To determine if a number is an evil number, you count the number of 1's in its binary representation (its binary digits), and if the count is even, then the number is evil. If the count is odd, then the number is called an odious number.


Here are a few examples of evil numbers:

  1. 3 is an evil number because its binary representation is 11, which has 2 1's, an even number of 1's.

  2. 6 is an evil number because its binary representation is 110, which has 2 1's, an even number of 1's.

  3. 10 is an evil number because its binary representation is 1010, which has 2 1's, an even number of 1's.

As you can see, evil numbers are named as such because their binary representation has an even number of 1's. Evil numbers can be used in a variety of math problems, including programming and computer science. They are a great way to introduce children to the concept of binary numbers and to help them understand how binary numbers work.


Pronic numbers

To determine if a number is a pronic number, you check if it can be expressed as the product of two consecutive integers. Here are a few examples of pronic numbers:

  1. 2 is a pronic number because it can be expressed as 1 × 2.

  2. 12 is a pronic number because it can be expressed as 3 × 4.

  3. 42 is a pronic number because it can be expressed as 6 × 7.

As you can see, pronic numbers are named as such because they are the product of two consecutive integers.


Repunit numbers

To determine if a number is a repunit number, you check if all its digits are 1's. Here are a few examples of repunit numbers:

  1. 1 is a repunit number because it has one digit and that digit is 1.

  2. 11 is a repunit number because it has two digits and both digits are 1's.

  3. 111 is a repunit number because it has three digits and all three digits are 1's.

As you can see, repunit numbers are named as such because they are composed entirely of repeated 1's. Repunit numbers are used to study prime numbers and to find patterns in their distribution.




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