Belphegor’s Prime is a specific palindromic prime number that contains the numerical sequence 666, flanked by thirteen zeros on each side and capped by the digit 1 at both ends. The number is written as 1,000,000,000,000,066,600,000,000,000,001. In scientific notation, this 31-digit integer is expressed as $10^{30} + 666 \times 10^{14} + 1$.

Beyond its mathematical properties, this prime number has gained notoriety in the field of recreational mathematics due to its association with superstitious symbols—the number 666, often referred to as the "Number of the Beast," and the number 13, widely considered unlucky in Western culture.

Defining the Anatomy of Belphegor’s Prime

To understand why Belphegor’s Prime stands out among the infinite sea of prime numbers, one must examine its symmetrical structure. A palindromic prime is a prime number that is also a palindrome, meaning it reads the same forward and backward. While primes like 11, 131, and 151 are common examples, Belphegor’s Prime is much larger and carries a more complex internal arrangement.

The breakdown of the 31 digits is as follows:

  • The Boundaries: It begins and ends with the digit 1.
  • The Zeros: There are exactly 13 zeros positioned immediately after the first 1 and another 13 zeros immediately before the final 1.
  • The Core: The heart of the number is the sequence 666.

The total length of the number is 31 digits. Interestingly, 31 is the reverse of 13, further contributing to the numerical coincidences that enthusiasts find fascinating. In the short scale naming convention (common in the United States), the number is read as one nonillion, sixty-six quadrillion, six hundred trillion, and one.

The Discovery by Harvey Dubner

The existence of this prime was first brought to light by Harvey Dubner, a mathematician and electrical engineer known for his contributions to finding large prime numbers. Dubner specialized in identifying primes with specific structures, often utilizing custom-built hardware to perform computations that were difficult for standard consumer computers of his time.

In the mid-1990s, Dubner was exploring a sequence of palindromic numbers that followed the form of $10^n$ variations. During his search, he identified that when $n=13$ in a specific configuration involving the digits 666, the resulting 31-digit number was prime. While the discovery was recorded in mathematical databases, it remained a niche observation within professional circles until it received a distinctive name that captured the public imagination.

Naming and the Demon of Ingenuity

The name "Belphegor’s Prime" was coined in 2012 by Clifford A. Pickover, an author and mathematician famous for making complex mathematical concepts accessible and engaging through storytelling. Pickover named the prime after Belphegor, one of the Seven Princes of Hell in Christian demonology.

According to lore, Belphegor is the demon of sloth, but he is also credited with tempting mortals by suggesting ingenious inventions and discoveries that lead to wealth. The choice of name was a deliberate attempt to link the mathematical properties of the number—specifically the "hellish" 666 and the "unlucky" 13—with the mythological figure. This branding transformed a dry mathematical fact into a cultural phenomenon, leading to its appearance in television shows like Elementary and various digital media platforms.

The Mathematical Sequence of Belphegor Numbers

Belphegor’s Prime is not an isolated anomaly but rather the second member of a larger family of palindromic numbers known as Belphegor numbers. A Belphegor number is defined by the general form:

$$B_n = 10^{2n+4} + 666 \times 10^{n+1} + 1$$

In this formula, $n$ represents the number of zeros flanking the central 666.

  • When $n=0$, the number is 16,661. This is also a prime number and is considered the first Belphegor Prime.
  • When $n=13$, we get the famous 31-digit Belphegor’s Prime.
  • When $n=42$, the number contains 42 zeros on each side of the 666. This is the third prime in the sequence.

The sequence of $n$ values that result in prime numbers is archived in the Online Encyclopedia of Integer Sequences (OEIS) under the designation A232448. Current mathematical records indicate that primes occur when $n$ equals 0, 13, 42, 506, 608, 2472, and 2623. As $n$ increases, the numbers become gargantuan, requiring significant computational power to verify their primality.

Primality Testing for Large Palindromes

Determining whether a 31-digit number like Belphegor’s Prime is truly prime requires more than simple trial division. For a number of this size, mathematicians use primality tests that can determine "primeness" without finding every factor.

The Miller-Rabin Test

One of the most common algorithms used in the discovery of such primes is the Miller-Rabin primality test. This is a probabilistic test; if a number fails the test, it is certainly composite. If it passes multiple rounds of the test, the probability that it is prime becomes extremely high—so high that for all practical purposes, it is accepted as prime.

Deterministic Tests

To be 100% certain, deterministic tests like the Baillie-PSW primality test or the Elliptic Curve Primality Proving (ECPP) method are used. For Belphegor’s Prime, which is relatively small by modern cryptographic standards (which often use primes with hundreds or thousands of digits), these tests can confirm its status in a fraction of a second on a modern processor.

Why 666 and 13 Matter in Recreational Mathematics

The fascination with Belphegor’s Prime highlights a branch of study known as recreational mathematics. This field focuses on the aesthetic, surprising, and cultural aspects of numbers rather than just their utility in physics or engineering.

The inclusion of the "Number of the Beast" (666) is a significant draw. In the Book of Revelation, 666 is associated with an apocalyptic creature. In mathematics, 666 is actually quite interesting on its own:

  • It is the sum of the squares of the first seven prime numbers ($2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666$).
  • It is a triangular number, specifically the sum of the integers from 1 to 36.

Similarly, the number 13 has a long history of "unluckiness" in Western cultures, a phenomenon known as triskaidekaphobia. The presence of 13 zeros in Belphegor’s Prime is the primary reason it is considered "spooky" or "evil." When combined with the fact that the total digit count (31) is the mirror image of 13, the number becomes a perfect subject for mathematical folklore.

Computational Challenges of the Belphegor Sequence

While the $n=13$ case is well-documented, searching for higher-order Belphegor primes presents a significant challenge for distributed computing projects. As the number of zeros ($n$) increases, the digit count grows at a rate of $2n+7$.

For example, when $n=2623$, the resulting prime has over 5,000 digits. Testing these candidates requires specialized software like PFGW (Prime Form Group Walker) or LLR (Lucas-Lehmer-Riesel). These programs utilize Fast Fourier Transforms (FFT) to speed up the multiplication of massive integers, allowing researchers to scan thousands of candidates for primality.

The search for these numbers is part of a broader interest in "Prime Curios." These are numbers that are not necessarily useful for encryption (like RSA primes) but are prized for their unique patterns. Palindromic primes, repunit primes (consisting only of the digit 1), and Mersenne primes all fall into this category of mathematical exploration.

The Role of Belphegor’s Prime in Modern Culture

The digital age has allowed Belphegor’s Prime to transition from a footnote in a mathematical journal to a viral topic. It frequently appears in "dark academia" aesthetics and internet mystery circles. Clifford Pickover’s playful warning to "not stare at the number for too long" added a layer of creepypasta-like intrigue that helped the number reach an audience far beyond mathematicians.

In the TV show Elementary, the number was used as a plot point involving a code, showcasing how writers leverage real mathematical curiosities to add a sense of realism and mystery to fiction. This intersection of hard science and pop culture is where Belphegor’s Prime truly thrives.

Mathematical Comparison with Other Famous Primes

To put Belphegor’s Prime in perspective, it is helpful to compare it to other famous prime types:

  1. Mersenne Primes: Numbers of the form $2^p - 1$. These are the largest known primes and are much bigger than Belphegor’s Prime, but they lack the palindromic symmetry.
  2. Sophie Germain Primes: A prime $p$ where $2p + 1$ is also prime. These are useful in cryptography and number theory but don't have the "visual" appeal of Belphegor's number.
  3. 73 (The Sheldon Prime): Popularized by The Big Bang Theory, 73 is the 21st prime number, and its mirror (37) is the 12th prime (the mirror of 21). While 73 is elegant, it lacks the "demonic" branding of Belphegor’s Prime.

Belphegor’s Prime occupies a unique niche because its "value" is derived almost entirely from its base-10 representation. If we were to calculate primes in base-12 or base-16, the number 1,000,000,000,000,066,600,000,000,000,001 would no longer appear as a palindrome with 666 at its center. It is a "base-dependent" curiosity.

The Symbolism of the Upside-Down Pi

In many graphical representations of Belphegor’s Prime, a specific symbol is used: an inverted Greek letter Pi ($\pi$). This symbol was first identified in the mysterious Voynich Manuscript, a 15th-century codex that remains undeciphered. Clifford Pickover adopted this symbol to represent the prime, further deepening the sense of historical mystery surrounding the number. While there is no direct mathematical link between the Voynich Manuscript and the prime number (which was discovered centuries later), the use of the symbol serves as a powerful visual mnemonic for the number's "arcane" nature.

Conclusion

Belphegor’s Prime is a fascinating intersection of number theory, computer science, and human psychology. Mathematically, it is a robust example of a palindromic prime that belongs to a specific, identifiable sequence ($B_{13}$). Culturally, it serves as a testament to the power of naming and symbolism. By taking an obscure 31-digit prime and layering it with the mythology of Belphegor, the "evil" of 666, and the "bad luck" of 13, mathematicians have created a doorway for the general public to enter the world of complex integers. Whether viewed as a demonic omen or a beautiful piece of numerical symmetry, the number remains one of the most recognizable curiosities in modern mathematics.

Summary

Belphegor’s Prime is the palindromic prime number 1,000,000,000,000,066,600,000,000,000,001. It is defined by its 31-digit length, featuring the "Number of the Beast" (666) centered between two sets of 13 zeros. Discovered by Harvey Dubner and named by Clifford Pickover, it belongs to a sequence of primes generated by the formula $10^{2n+4} + 666 \times 10^{n+1} + 1$. Its fame arises from the convergence of mathematical rarity and powerful cultural superstitions.

FAQ

What is the exact value of Belphegor’s Prime?

The exact value is 1,000,000,000,000,066,600,000,000,000,001.

Why is it called Belphegor’s Prime?

It was named by Clifford Pickover after Belphegor, one of the seven princes of hell, due to the number's inclusion of 666 and 13, which are traditionally associated with the occult or bad luck.

How many zeros are in Belphegor’s Prime?

There are 26 zeros in total: 13 on the left side of the 666 and 13 on the right side.

Is Belphegor’s Prime the only one of its kind?

No, it is part of a sequence. Other Belphegor primes occur when there are 0 zeros (16661) or 42 zeros on each side, among other rarer instances.

How was Belphegor’s Prime discovered?

It was discovered by mathematician Harvey Dubner using computational primality testing in the 1990s as part of his research into large palindromic primes.

What is the symbol for Belphegor’s Prime?

It is often represented by an upside-down Greek letter Pi ($\pi$), a symbol borrowed from the Voynich Manuscript.