Numbers play a crucial role in all aspects of human life. The invention and naming of numbers follow some fascinating rules.
Let’s explore interesting facts about numbers from a completely new perspective to uncover exciting discoveries in mathematics and numerology. However, while reading, please don’t try to understand everything if you are not genuinely curious, as some of these concepts may be quite mind-boggling.
1. Friendly Numbers
Two numbers form a pair of friendly numbers when they satisfy the rule: one number equals the sum of all the divisors of the other number (excluding the number itself), and vice versa. The first known pair of “friendly numbers” is 220 and 284. Let’s analyze this a bit: The number 220, apart from itself, has 11 divisors: 1, 2, 4, 5, 10, 11, 20, 44, 55, and 110. The sum of these 11 divisors equals 284. Conversely, the number 284 has 5 other divisors: 1, 2, 4, 71, and 142, which add up to 220.
In the 17th century, the French mathematician Fermat discovered the second pair of “friendly numbers”: 17296 and 18416. Around the same time, another French mathematician found the third pair: 9363544 and 9437056. The most astonishing revelation came in 1750 when the famous Swiss mathematician Leonhard Euler announced 60 pairs of friendly numbers at once, leaving the mathematical community in shock, believing “Euler had discovered them all.” However, a century later, a 16-year-old Italian named Baconi announced a new pair of friendly numbers in 1866, just slightly larger than 220 and 284: 1184 and 1210. Great mathematicians before him had overlooked this simple pair.
With the advancement of technology, mathematicians using computers have checked all numbers up to 1,000,000, ultimately finding 42 pairs of friendly numbers. Today, the number of known pairs of friendly numbers has exceeded 1,000. However, is there an infinite number of friendly numbers? Do they follow a specific distribution? These questions remain unanswered.
In the current technological age, with just a C++ algorithm that is not overly complex, you can find countless pairs of friendly numbers.
2. Amicable Numbers
Not stopping at friendly numbers, scientists have taken a step further to define “amicable numbers.”
An amicable pair consists of two positive integers such that the sum of the divisors of one number (excluding itself) exceeds the other number by exactly 1. In other words, (m, n) is an amicable pair if s(m) = n + 1 and s(n) = m + 1, where s(n) represents the sum of the proper divisors of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ represents the sum of divisors function.
The first known amicable pairs are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).
It has been proven that amicable pairs always consist of one even number and one odd number (perhaps symbolizing one male and one female).
3. Emirp
If you are trying to search for this term in English, you might not find it, as it is the word “Prime” spelled backward.
An emirp is a prime number that, when its digits are reversed, also yields a prime number. This definition excludes palindromic primes (like 151 or 787) and single-digit primes such as 7.
The first emirps discovered are: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157…
As of November 2009, the largest known emirp is 1,010,006,941,992,101 × 104,999,1, discovered by Jens Kruse Andersen in October 2007.
4. Perfect Numbers
In number theory, a positive integer is called a perfect number if it equals the sum of all its positive divisors, excluding itself. Alternatively, a number is termed perfect when it equals half the sum of its positive divisors (including itself). For example, the first perfect number is 6, as: 6 = 1 + 2 + 3, or 6 = (1 + 2 + 3 + 6)/2.
Historically, the first four perfect numbers: 6, 28, 496, and 8128 have been known since ancient Greek mathematics, discovered by the mathematician Nicomachus in the form: 2n−1(2n − 1):
- When n = 2: 21(22 − 1) = 6
- When n = 3: 22(23 − 1) = 28
- When n = 5: 24(25 − 1) = 496
- When n = 7: 26(27 − 1) = 8128.
Note that 2n − 1 is prime in each of the above examples. Euclid proved that the formula 2n−1(2n − 1) will yield an even perfect number if and only if 2n − 1 is prime (Mersenne prime).
In a manuscript written between 1456 and 1461, an anonymous mathematician introduced the fifth perfect number: 33,550,336. In 1588, the Italian mathematician Pietro Cataldi identified (8589869056) and (137,438,691,328) as the sixth and seventh perfect numbers.
Euclid proved that 2n−1(2n − 1) is a perfect number when 2p−1 is prime. For 2n − 1 to be prime, n must also be a prime number. For example: n = 2 => 2*(22−1) = 6; n= 3 => 22 (23−1) = 28. Prime numbers of the form 2n−1 are called Mersenne primes, named after the monk Marin Mersenne, who studied number theory and perfect numbers. By the 18th century, Leonhard Euler had proved that “every Mersenne prime generates a perfect number, and vice versa, every perfect number corresponds to a Mersenne prime.” This result is commonly referred to as the Euclid-Euler Theorem.
As of February 2013, 48 Mersenne primes and thus 48 perfect numbers have been discovered. The largest of these is 257,885,160 × (257,885,161−1) with 34,850,340 digits.
5. Powerful Numbers
The origin of this name comes from the tale of Achilles’ heel. As a powerful war hero, Achilles had only one weak point: his heel. Perhaps from here, the distinction between three terms arose: perfect numbers, Achilles numbers, and powerful numbers.
A number is called a powerful number if it is divisible by a prime number and also divisible by the square of that prime number. For example, the number 25 is a powerful number because it is divisible by the prime number 5 and the square of 5 (which is 25). Thus, a powerful number can also coincide with a perfect number (as defined above).
An Achilles number is a powerful number but not a perfect number.
Here is a list of all powerful numbers between 1 and 1000: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.
6. Quirky Numbers
To understand what quirky numbers are, we need to go through two definitions: Abundant Numbers and Semiperfect Numbers.
Abundant numbers are numbers for which the sum of their divisors (excluding themselves) is greater than the number itself. For example, the number 12 has divisors summing to (excluding 12) 1 + 2 + 3 + 4 + 6 = 16 > 12. Therefore, 12 is an abundant number.
Semiperfect numbers are natural numbers that equal the sum of all or some of their divisors. Thus, the set of semiperfect numbers is broader than the set of perfect numbers. Some examples of semiperfect numbers include: 6, 12, 18, 20, 24, 28, 30, 36, 40…
Therefore, there are common elements between the sets of semiperfect numbers and abundant numbers.
And finally, what is a quirky number? A number is quirky if it is an abundant number but not a semiperfect number. In other words, the sum of its divisors is greater than the number itself, but the sum of some or all of its divisors never equals the number itself.
Some of the first numbers in the set of quirky numbers include: 70, 836, 4030, and 5830.
7. Happy Numbers
A happy number is defined by the following process:
Starting with any positive integer, replace the number with the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will remain), or it enters an endless cycle that does not include 1.
Numbers that end this process at 1 are known as happy numbers, while those that do not end at 1 are referred to as unhappy numbers (or sad numbers).
Let’s try starting with the number 44:
+ First, 4 ^ 2 + 4 ^ 2 = 16 + 16 = 32.
+ Next: 3 ^ 2 + 2 ^ 2 = 9 + 4 = 13.
+ And again: 1 ^ 2 + 3 ^ 2 = 1 + 9 = 10.
+ Finally: 1 ^ 2 + 0 ^ 2 = 1 + 0 = 1.
That’s a happy number.
Interestingly, happy numbers are quite common, with 143 happy numbers between 0 and 1000. The largest happy number without any repeating digits is: 986,543,210. That is indeed a happy number.
8. Unabbreviated Numbers
This quirky name is given to numbers that “cannot” be expressed as the sum of all the divisors of any positive integer (excluding the integer itself).
For example, 4 is not an unabbreviated number because 4 = 3 + 1. Here, 3 and 1 are all the divisors of 9. On the other hand, 5 is an unabbreviated number because the only way to express it is 5 = 4 + 1. If you argue that this is the sum of the divisors of 4, you are mistaken. The sum of the divisors of 4 must be: 1 + 2 = 3.
The first unabbreviated numbers are: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290…
9. Narcissistic Numbers
Narcissistic numbers are those that are equal to the sum of the cubes of their own digits. For example:
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.
These numbers, when named by scientists, recognize their own vanity. British mathematician G.H. Hardy even published in his book “A Mathematician’s Apology”: “These are strange concepts, very suitable for puzzle columns and capable of entertaining, but they hold no appeal for mathematicians.” Nevertheless, it presents readers with a new perspective on mathematics.
