Belphegor's Prime

There's a whole episode about this in TV Series - Elementary and it has more concepts like this. I honestly loved them.

Belphegor's prime is the palindromic prime number 1000000000000066600000000000001 (1030 + 666 × 1014 + 1), a number which reads the same both backwards and forwards and is only divisible by itself and one.

Simple explanation

Belphegor's Prime is a famous curiosity in mathematics.

It is a number that looks like this:

1,000,000,000,000,066,600,000,000,000,000,1

If you remove the commas, it has:

a 1
thirteen 0s
666
thirteen 0s
another 1

What makes it special is that mathematicians checked it and found that it is a prime number—a number divisible only by 1 and itself.

The fascination is not that it's useful. The fascination is that such a strangely patterned number happens to be prime.

The core idea

Belphegor's Prime is a deliberately constructed number with a striking pattern:

1 + thirteen zeros + 666 + thirteen zeros + 1

The surprise is that despite looking designed and highly structured, it is still a prime number.

Key concepts

Prime number

A number is prime if its only positive divisors are 1 and itself.

The pattern is the attraction

Most famous primes are famous because of mathematical properties. Belphegor's Prime is famous because of its appearance.

The "666" is intentional

The creator placed 666 in the middle because of its cultural association as the "number of the beast."

The name comes from Belphegor

Belphegor is a demon figure from European folklore. The name was chosen because of the 666 in the middle.

Thirteen zeros on each side

The number uses thirteen zeros before and after 666, creating a highly symmetric structure.

Not every similar pattern is prime

If you change the number of zeros, many versions become composite (non-prime).

It's recreational mathematics

This belongs to the world of mathematical curiosities and puzzles, not practical engineering or finance.

One analogy

Imagine a safe manufacturer creates a key that looks absurdly decorative—perfect symmetry, weird symbols, obvious patterns.

You would expect it to be easy to copy or break.

Then you discover it opens a lock that nobody can crack.

Belphegor's Prime creates the same feeling: a number that looks almost too designed to be mathematically special, yet it turns out to be prime.

Where it actually matters

Understanding what "prime" really means

Many people subconsciously think prime numbers should look random. Belphegor's Prime shows that appearance tells you almost nothing.

Learning mathematical proof vs intuition

Your intuition says, "This looks suspiciously constructed, so it probably has factors." Mathematics says, "Check it."

Recreational mathematics

It's often used to spark interest in number theory because the pattern is memorable.

Thinking about large numbers

It demonstrates how difficult primality can be. Looking at a huge number rarely tells you whether it's prime.

Common confusions

"Because it contains 666, the number has some hidden mathematical significance."

Wrong model: The 666 causes the prime property.

Why it forms: The pattern is so striking that people assume the digits themselves create the primality.

In reality, 666 is just part of the decoration.

"A symmetric number is more likely to be prime."

Wrong model: Beautiful structure implies mathematical rarity.

Why it forms: Humans are pattern-seeking. We tend to associate symmetry with special mathematical properties.

Most patterned numbers are not prime.

"Belphegor's Prime was discovered naturally."

Wrong model: Someone found it while studying deep number theory.

Why it forms: Many famous primes arise from research.

Belphegor's Prime was intentionally constructed as a mathematical curiosity.

Test your understanding

If the 666 were replaced by 777, would you expect the resulting number to automatically be prime? Why or why not?
What is the surprising fact about Belphegor's Prime: its digit pattern or its primality?
Why can't you determine whether a huge number is prime merely by looking at its digits?

Q&A (5 pairs)

Why is Belphegor's Prime memorable among thousands of known primes? — Unusual symmetric digit pattern.

Does digit symmetry imply primality? — No, almost never.

Was the 666 mathematically required? — No, purely aesthetic.

Can similar-looking numbers be composite? — Yes, most are.

Is Belphegor's Prime theoretically important? — Mostly recreational mathematics.

Accuracy note

This explanation focuses on the famous recreational-math version of Belphegor's Prime and intentionally skips the technical details of how its primality was verified. For professional number theory, the interesting part is not the folklore or the 666—it is the computational proof that the number has no non-trivial factors. The cultural references (Belphegor, 666, thirteen zeros) are historical naming choices, not mathematical reasons for the number being prime.

Canva Slides for Belphegor's Prime