Buzz
Mathematicians take pleasure in categorizing numbers in numerous ways. Natural numbers are fundamental for counting and ordering; nominal numbers are used for identification (like a driver's license number); integers are whole numbers that don't involve fractions or decimals; prime numbers are those that can only be divided by 1 and themselves, and so on. The potential ways to understand and use numbers are limitless, which is why a branch of mathematics called ‘number theory’ focuses on the study of integers. Although we now know number theory has endless applications and uses, it can often seem trivial, particularly the aspect known as ‘recreational number theory.’ As number theorist Leonard Dickson once said, 'Thank God that number theory is unsullied by any application.'
That being said, this doesn’t mean it isn’t a source of geeky enjoyment for those who are interested. Keep reading to discover what makes a number 'interesting,' 'weird,' 'happy,' 'narcissistic,' 'perfect,' and more!
10. Amicable Numbers
Ah, amicable numbers. They share such a special bond. Just how strong is their connection? Let's examine a classic pair—284 and 220—and see just how close they are. We begin by adding all the proper divisors of 220 (those divisors that result in no remainder, including 1, but excluding the number itself):
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
Next, let’s do the same thing with 284:
1 + 2 + 4 + 71 + 142 = 220.
And there we have it: a perfect pair of amicable numbers. Other examples include (1184, 1210), (2620, 2924), and (5020, 5564). This fascinating number pair was first discovered and studied by the Pythagoreans, and has been the subject of considerable research over the centuries. Prominent mathematicians such as Fermat, Descartes, Iranian Muhammad Baqir Yazdi, and Iraqi Th?bit ibn Qurra have explored the nature of amicable numbers. Ongoing studies focus on determining whether an infinite number of pairs exist, identifying patterns, and understanding the reasons behind this unique phenomenon.
Because mathematicians are never content with just amicable numbers, they’ve also introduced “betrothed numbers”—pairs in which the sum of the proper divisors of each number equals the other number plus 1.
9. Emirp
The term ‘Emirp’ is simply the word ‘prime’ spelled backwards, and refers to a prime number that transforms into another prime number when its digits are reversed. Emirps do not include palindromic primes (such as 151 or 787) or single-digit primes like 7. Some of the first emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, and 157 – reverse these, and you get a new prime each time.
Honestly, just saying ‘emirp’ repeatedly is kind of fun. Give it a try!
8. Interesting Numbers
There exists a classic paradox in mathematics known as the 'interesting number paradox.' In simple terms, as you keep counting natural numbers, you'll eventually come across one that seems uninteresting. But the paradox arises when you realize that, by being the smallest uninteresting number, this number has, in fact, become interesting.
Of course, this concept is entirely subjective, as it depends on an unclear definition of what 'interesting' means. Broadly speaking, a number is deemed interesting if it has some special mathematical property. For example, 19 is interesting because it’s prime, 999 is interesting due to its status as a palindrome (and also the UK version of 911), and 24 is interesting because, among other reasons, it is the largest number that can be evenly divided by all numbers smaller than its square root. Mathematicians
7. Powerful Numbers
Achilles was a mighty hero of the Trojan War, known for his incredible strength but also for his one vulnerability—his Achilles' heel. Similarly, Achilles numbers are powerful, yet they’re not flawless.
Let’s start with a powerful number. A number is deemed powerful if all of its prime factors continue to be factors when squared. For example, 25 is a powerful number because its only prime factor, 5, remains a factor even after squaring it (25, which divides 25 evenly). Now, let's shift to perfect powers, numbers that can be expressed as an integer raised to another integer's power; for example, 8 is a perfect power because it’s 2 cubed.
Returning to our original idea, Achilles numbers are powerful but not perfect powers. The first Achilles number is 72, as it is powerful but not a perfect prime. Other examples include 108, 200, 288, 392, 432, 500, and 648.
6. Weird Numbers
What exactly are weird numbers? To grasp them, we first need to look at “abundant” numbers. These are numbers that are greater than the sum of their proper divisors. For example, 12 is the smallest abundant number because the sum of its proper divisors (1+2+3+4+6) equals 16. So, 12 has an “abundance” of 4, the amount by which the sum of its divisors exceeds the number itself. While many abundant numbers are even, we don’t encounter an odd one until we reach 945.
Some abundant numbers are “semiperfect” or “pseudoperfect,” meaning they are equal to all or some of their proper divisors. 12, for instance, is an imperfect abundant number because its divisors can be added together to equal 12.
Now we come to weird numbers. A number is considered weird if it is abundant but NOT semiperfect. In other words, the sum of its divisors exceeds the number, but no subset of those divisors adds up to the number itself. Weird numbers are rare—the first few examples include 70, 836, 4,030, and 5,830.
5. Untouchable Numbers
While weird numbers cannot be the sum of any of their divisors, untouchable numbers go even further. For a number to be untouchable, it must not be the sum of the proper divisors of ANY number. Some examples of untouchable numbers are 2, 5, 52, and 88; interestingly, 5 is believed to be the only odd untouchable number, though this has not been formally proven. There is an infinite number of untouchable numbers, so no such thing as the largest one exists.
4. Perfect Numbers
After exploring weird and untouchable numbers, we now turn to the ultimate category of proper divisor-related numbers: perfect numbers. A perfect number is one that is exactly equal to the sum of its proper divisors (excluding the number itself). The first perfect number is 6, as the sum of its divisors (1, 2, 3) equals 6. Following 6, the next perfect numbers are 28, 496, and 8,128. Ancient Greek mathematicians were aware of these first four perfect numbers, with Nichomatus discovering 8,128 around A.D. 100. Three more were uncovered later, with the first in 1456 (33,550,336) by an unknown mathematician, and two more in 1588 (8,589,869,056 and 137,438,691,328) by the Italian mathematician Pietro Cataldi.
All known perfect numbers are even, and it remains an open question whether any odd perfect number exists, or if it is even possible. English mathematician James Joseph Sylvester once wrote, “...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.”
3. Repdigits and Reunits
A repdigit is a natural number made up of a single repeating digit. The name itself comes from “repeated digit.” The most infamous repdigit is 666, also known as the “Beast Number,” commonly associated with the antichrist or Satan. A repunit, on the other hand, is a repdigit that exclusively uses the number 1. Repunits appear often in binary code and are closely tied to Mersenne Primes, one of the most famous families of primes. It is hypothesized that there are infinitely many repunit primes, so if you're up for the challenge, feel free to try and prove it yourself.
2. Narcissistic Numbers
Narcissistic numbers, also referred to as Armstrong numbers or “pluperfect digital invariants,” are numbers that are equal to the sum of their own digits when each digit is raised to the power of the total number of digits in the number.
Confused? Let’s clear things up with an example using the four well-known narcissistic cubes:
153 = 1^3 + 5^3 + 3^3 370 = 3^3 + 7^3 + 0^3 371 = 3^3 + 7^3 + 1^3 407 = 4^3 + 0^3 + 7^3
In these examples, each digit is cubed because the numbers have three digits. Then, the cubes of these digits are summed up to equal the original number. There are no 1-digit narcissistic numbers, nor any 12 or 13-digit ones; however, there are two 39-digit narcissistic numbers, which are:
115132219018763992565095597973971522400 and 115132219018763992565095597973971522401.
English mathematician G. H. Hardy dismissed the significance of such numbers in his book “The Mathematician’s Apology,” stating, “These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.”
1. Happy Numbers
Some numbers are odd; others are happy. If you want to determine whether a number is happy, follow this sequence of steps. Let’s take the number 44 as an example:
First, square each of the digits, then sum them together:
4^2 + 4^2 = 16 + 16 = 32
Next, we repeat the process with our new number:
3^2 + 2^2 = 9 + 4 = 13
Now, let's do it again:
1^2 + 3^2 = 1 + 9 = 10
And at last:
1^2 + 0^2 = 1 + 0 = 1
Voila! It's a happy number. Whenever you apply this method to any number and eventually land on 1, you’ve found a happy number. If it never reaches 1, unfortunately, it's not a happy number. Interestingly, happy numbers are quite common; for example, there are 11 of them between 1 and 50.
As a final tidbit, the largest happy number without any repeating digits is 986,543,210. Now, that's truly a happy number.
