Prime numbers so memorable that people hunt for them

Original Article ↗ Hacker News Discussion ↗

Prime numbers with striking digit patterns—from “Belphegor’s prime” framed by 666 to palindromic and image-forming primes—are inspiring people to hunt for them as mathematical curiosities and even forms of digital art. Commenters connect these “memorable” primes to ideas like the interesting number paradox, phone-number and word-based primes, and base-dependent notions of what looks special. Some challenge claims that such primes are useful in cryptography, arguing that real-world schemes require large, secret or carefully chosen primes rather than aesthetically pleasing ones, and note how tools like LLMs still struggle with reliably identifying large primes.

Memorable and “Artistic” Primes

  • Discussion centers on primes with visually or culturally striking patterns: Belphegor’s prime (1 + zeros + 666 + zeros + 1), Trinity Hall prime, “PPcg prime,” Christmas-tree primes, and the “HN prime” rendered as a bitmap in binary.
  • People note that families of such primes exist (e.g., Belphegor primes with varying zero counts) and that many are cataloged in OEIS.
  • Several participants link this to the idea of hunting for primes that encode images or text in their digit patterns.

Palindromes and “Interesting” Numbers

  • Multiple comments explore palindromic primes, including facts like: any palindrome with an even number of decimal digits (beyond 11) is divisible by 11.
  • Examples of palindromic or pattern-based primes (e.g., 3,212,123; powers of repeated 1s; 111…1² giving 123…n…321) are shared.
  • The “interesting number paradox” and the taxicab number anecdote are used to argue that every integer can be considered interesting in some way.

Base Choice and Non-Decimal Representations

  • Some argue base 10 is an evolutionary accident and suggest searching in other bases (binary, base‑7, base‑11, base‑36).
  • Binary palindromes and primes that spell words in base‑36 (e.g., “did”, “nun”, “primetest”) are mentioned.
  • There’s experimentation with mapping letters to digits and checking which palindromic words yield prime numbers.

Anecdotes About Specific Numbers

  • The “Grothendieck prime” story (choosing 57 as a “prime”) is discussed as illustrating extreme abstraction from concrete numbers.
  • Debate over which composite feels “most prime” (e.g., 57, 87, 91), with references to arguments that 91 “wins.”
  • Personal stories include prime and palindromic phone numbers and factoring odometer readings for fun.

Cryptography and Memorable Primes

  • Some skepticism is expressed about the article’s claim that memorable primes help cryptography; RSA needs secret primes, so memorability is seen as a liability.
  • Others note public, structured primes are important as domain parameters in systems like elliptic-curve cryptography; examples include primes of the form 2^n − k (e.g., “25519”).

LLMs, Math, and OEIS

  • A shared example shows an LLM confidently misclassifying a large patterned number as composite despite code access.
  • This leads to calls for tighter integration of LLMs with symbolic math tools and databases like OEIS, and debate over whether “LLMs can’t do math” is a fair criticism.