Equidigital, Extravagant, and Frugal Numbers

Equidigital, Extravagant, and Frugal Numbers are all related in that they deal with the number of digits in their prime factorization and the nummber of digits of the number itself.


Extravagant numbers are the most numerous, for there are infinitely many of them. We are starting with extravagant numbers because they may be the simplest to understand and they encompass the vast majority of real, natural numbers, plus understanding extravagant numbers will help you understand equidigitals and frugal numbers. Extravagant numbers are also called wasteful numbers because they are natural numbers that have fewer digits themselves than the number of digits in their prime factorization. Think of them as wasteful in that they may have many large factors, but they waste their larger composite factors because those must be broken down into prime factors. Most, but not all composite numbers will be extravagant numbers. All extravagant numbers are composite because they are factorable, so, there are no prime extravagant numbers. The first few extravagant numbers are 4, 6, 8, 9, and 12, but not 10! To find out if a composite, natural number is extravagant, first find its prime factors:


4 = 2 * 2 = 2^2
6 = 2 * 3
8 = 2 * 2 *2 = 2^3
9 = 3 * 3 = 3^2
12 = 4 * 3 or 6 * 2 = 2 * 2 * 3


In the case of 4, 6, 8, and 9, they each have one digit themselves and are composed of two or more prime factors, making them extravagant. See, 12 may have 4 and 6 as factors, but they are composite and must be broken down further into prime factors. The number 12's prime factorization is 2 * 2 * 3, which makes three digits, 12 only has two digits, so it is extravagant. The composite number 10 is not extravagant though. Read on to Equidigital and Frugal Numbers to find out more...


Equidigital numbers are the next most numerous numbers because this set of numbers encompasses all prime numbers and many of the left over composite numbers that arent wasteful. Equidigital numbers are also known as economical numbers because they do not have to break down large composite factors (ie all their factors are prime already) and they have the same (equal) number of digits as does the number of digits in  their prime factorization. The first few equidigital numbers are 1, 2, 3, 5, 7, and 10. Take a look at their prime factors:


1 = 1
2 = 2
3 = 3
5 = 5
7= 7
10 = 2 * 5

1 is unique for it is neither prime nor composite, but a 'unit'. 1's prime factorization by definition is 1. The numbers 2, 3, 5, and 7 are prime, so they have already reached prime factorization, each one digit like the numbers themselves. Ten however is two digits in length and can be factored into 2 and 5, which makes two digits, so 10 is equidigital too! Read onward to find out about Frugal Numbers!

Frugal numbers, as you could assume by now are the least numerous set of numbers because this set of numbers more or less picks up what is left over from the sets of extravagant and equidigital numbers. You could also assume by now that frugal numbers are also known as economical numbers because they are natural numbers that do not have to break down large composite factors (again, all their factors are prime already), just like equidigital numbers. However, unlike equidigitals, frugal numbers have more digits by themselves than the number of digits in their prime factorization. There are no single digit or double digit frugal numbers and only 8 triple digit frugal numbers. They are 125, 128, 243, 256, 343, 512, 625, and 729. After this, frugal numbers become a bit more numerous: 1024, 1029, 1215, 1250, 1280...

125 = 5 * 5 * 5 = 5^3
128 = 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7

Examples/Review:
Identify whether the following number is Equidigital (economical), Extravagant (wasteful), or Frugal (also economical).
1. 64
2. 1024
3. 24
4. 42
5. 25
6. 125
7. 1215
8.111
9. 12
10. 10

Answers: 1.) Equidigital 2.) Frugal 3.) Extravagant 4.) Extravagant 5.) Equidigital 6.) Frugal 7.) Frugal 8.) Equidigital 9.) Extravagant 10.) Equidigital

-Zech