Exploring Mathematical Wonders of the Number 666

Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number.
G. H. Howison

Numbers are integral to mathematical discussions and play a major role in our lives, allowing us to quantify and create order. However, numbers can also be appreciated for their own sake, as they may possess unusual properties, exhibit hidden beauty, or simply evoke marvel at their inherent nature.

For instance, let’s consider the number 8. The uniqueness of 8 is subject to interpretation. Mathematically, 8 is a perfect cube: 8 = 2³. Additionally, it is the only cube that is one less than a perfect square, 9. Moreover, 8, being the sixth Fibonacci number, stands out as the sole Fibonacci number (aside from 1) that is a perfect cube, and so on.

This blog post explores the intriguing number 666. While widely known for its biblical association as the number of the beast and associated with bad luck omens, we will focus solely on its mathematical properties, many of which are remarkably fascinating.

To begin with, the number 666 is a palindrome (i.e., it reads the same in both directions).

It is the sum of two consecutive palindromic prime numbers:

666 = 313 + 353.

We can represent 666 in Roman numerals as 666 = DCLXVI, where D = 500, C = 100, L = 50, X = 10, V = 5, I = 1. Notice that all the numerals less than 1000 are used and in descending order!

The number 666 just happens to be the sum of the first thirty-six numbers:

1 + 2 + 3 + 4 + 5 + ··· + 34 + 35 + 36 = 666.

666 is the sum of the squares of first seven prime numbers 2, 3, 5, 7, 11, 13, and 17:

2² + 3² + 5² + 7² + 11² + 13² + 17²

= 4 + 9 + 25 + 49 + 121 + 169 + 289

= 666.

Another amazing fact is through its prime factors. We have 666 = 2 × 3 × 3 × 37. Sum of the digits of 666 is 6 + 6 + 6 = 18 is same as the sum of the (individual) digits of its prime factors 2 + 3 + 3 + 3 + 7 = 18.

The number 666 is equal to the sum of the cubes of the digits of its square, plus the digits of its cube. We have

666² = 443556  

666³ = 295408296

Sum of the cubes of the digits of the square of 666 is 4³ + 4³ + 3³ + 5³ + 5³ + 6³ = 621

Sum of the digits of cube of 666 is 2 + 9 + 5 + 4 + 0 + 8 + 2 + 9 + 6 = 45

Finally, as expected, the sum is

621 + 45 = 666

Using the digits of the number 666 and various mathematical operations, we have the following nice relation:

(6 × 6 × 6)² + (666 ‒ 6 × 6)² = 666²

We can write 666 using the first nine natural numbers in increasing order and then placing the + sign suitably, as follows:

1 + 2 + 3 + 4 + 567 + 89 = 666

123 + 456 + 78 + 9 = 666

Similarly, in decreasing order, we can write the same as

9 + 87 + 6 + 543 + 21 = 666

Here are a few surprising number relationships that lead to 666:

666 = 1⁶ ‒ 2⁶ + 3⁶

666 = (6 + 6 + 6) + (6³ + 6³ + 6³)

666 = (6⁴ ‒ 6⁴ + 6⁴) ‒ (6³ + 6³ + 6³) + (6 + 6 + 6)

666 = 5³ + 6³ + 7³ ‒ (6 + 6 + 6)

666 = 2¹ × 3² + 2³ × 3⁴

We can even generate 666 by representing each of its three digits in terms of 1, 2 and 3:

6 = 1 + 2 + 3

6 = 1 × 2 × 3

6 = √(1³ + 2³ + 3³).

Therefore, using the above relations

666 = (100)(1 + 2 + 3) + (10)(1 × 2 × 3) + √(1³ + 2³ + 3³).

666 is related to (6² + n²) in the following interesting ways:

666 = (6 + 6 + 6) · (6² + 1²)

666 = 6! · (6² + 1²) / (6² + 2²)

The sum of the numbers on a roulette wheel is 666.

The sum of the first 144 (= (6 + 6) · (6 + 6)) digits of π (pi) is 666.

A well-known approximation to π (pi) is 355/113 = 3.1415929… If one part of this fraction is reversed and added to the other part, we get 666:

553 + 113 = 666.

355 + 311 = 666

Remarkably, ϕ(666) = 6 × 6 × 6, where ϕ(n) is the Euler’s phi function.

A dottable fraction is a proper fraction where multiplication signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. Consider the following noteworthy dottable fraction consisting the number 666:

Our fascination with the number 666 is merely an exhibition of the beauty that lies within mathematics. Exploring the recreational side of mathematics is an enjoyable by-product of its crucial role in scientific exploration and discovery.

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call “Math1089 – Mathematics for All!“.