---
title: Math-labelled sats (Labs)
summary: A Labs experiment tagging each sat by the mathematical class of its number: prime, square, Fibonacci, perfect and so on. Our own labels, not ord's rarity.
published: 2026-10-08
updated: 2026-10-08
---
## In one sentence

Every sat has a number, from 0 to 2,099,999,997,689,999; this experiment labels that number by the mathematical classes it belongs to, in tiers from common to rare.

## These are our labels, not ord's rarity

ord's rarity (common, uncommon, rare, epic, legendary, mythic) comes from where a sat sits in Bitcoin's block subsidy schedule, and ord decides it. Nothing here changes it. A sat labelled `fibonacci` here is still whatever rarity ord gives it, and where our own ord index answers, the lookup shows both side by side.

## The labels

| Tier | Label | Rule |
| --- | --- | --- |
| Rare | `fibonacci` | The number is a Fibonacci number: 0, 1, 2, 3, 5, 8, 13, ... (74 below the total supply; 26 of them spendable). |
| Rare | `power_of_2` | The number is 2 to some power, 1 included (51; 18 spendable). |
| Rare | `factorial` | The number is k! for some k (17 distinct numbers, since 0! and 1! are both 1; 5 spendable). |
| Rare | `perfect` | The number equals the sum of its proper divisors, like 28 = 1 + 2 + 4 + 7 + 14 (7; 2 spendable). |
| Rare | `mersenne_prime` | The number is 2^p − 1 and prime (8; none spendable). |
| Rare | `highly_composite` | The number has more divisors than every smaller number, like 12, 360 or 720,720 (130; 56 spendable). |
| Rare | `primorial` | The number is the product of the first few primes: 1, 2, 6, 30, 210, ... (14; 4 spendable). |
| Rare | `factorial_prime` | A prime one more or one less than a factorial, like 719 = 6! − 1 (10; 1 spendable, 14! − 1). |
| Rare | `primorial_prime` | A prime one more or one less than a primorial, like 211 = 210 + 1 (12; 2 spendable). |
| Rare | `taxicab` | The smallest number that is a sum of two cubes in k different ways: 2, 1729 (Hardy and Ramanujan's taxi), 87,539,319 and 6,963,472,309,248 (4; 1 spendable). |
| Uncommon | `square` | The number is k × k (about 45.8 million). |
| Uncommon | `cube` | The number is k × k × k (about 128,000). |
| Pattern | `palindrome` | The decimal digits read the same both ways, like 1,234,321. |
| Pattern | `repdigit` | Two or more decimal digits, all the same, like 7,777. |
| Pattern | `armstrong` | The number equals the sum of its digits each raised to the number of digits, like 153 = 1³ + 5³ + 3³ (41 below the total supply). |
| Pattern | `automorphic` | The number's square ends in the number itself, like 76² = 5,776 (28; 9 spendable). |
| Common | `prime` | The number is prime (about 2.8% of all sats, roughly 60 trillion). |

A sat can carry several labels; its tier is the highest of them (rare, then uncommon, then pattern, then common). Where a definition could go either way, we follow the standard integer sequence for it (OEIS): 0 counts as a Fibonacci number, a square and a cube; 1 counts as 2⁰; single digits count as palindromes.

## How it's worked out

- **Primes:** a deterministic Miller-Rabin test with the first nine prime bases, which is a proof, not a probability, for every number below 3.8 × 10¹⁸; the largest sat number is about 2.1 × 10¹⁵.
- **Squares and cubes:** exact integer square and cube roots. No floating point, which can be off by one at this size.
- **Rare classes:** each list is generated in full when the dashboard starts (Fibonacci by addition, powers of 2 by doubling, factorials by multiplication, perfect numbers from Mersenne primes by the Euclid-Euler theorem). No odd perfect number exists below 10¹⁵⁰⁰, so the list is complete.
- **Armstrong numbers:** found by a search over every combination of digits, stored, and re-checked by the test suite.
- **Highly composite numbers:** every one is a product of the first primes with exponents that never increase (2³ × 3 × 5, not 2 × 3³), so only those numbers are checked, keeping each new record for the number of divisors.
- **Factorial and primorial primes:** the numbers either side of each factorial and primorial, tested for primality as above.
- **Automorphic numbers:** for each number of digits there are exactly four candidates, worked out directly, not searched for.
- **Taxicab numbers:** stored, because proving 6,963,472,309,248 is the *smallest* number with four cube-pair sums is a long search (published by Rosenstiel, Dardis and Rosenstiel in 1991). We check that it does have four, and our tests confirm the first three by search. The next one is far beyond the last sat.
- **Ranges:** to find labelled sats inside a range (a wallet's UTXO, say), the rare lists are searched directly, squares and cubes come from integer roots, and palindromes are built from their first half; nothing scans the range sat by sat. Primes are never listed over a range, only estimated with the logarithmic integral.

## Spendable or not

The genesis block's 50 BTC can never be spent, so sats 0 to 4,999,999,999 can never move. Any result in that range says `unspendable`. Most of the famous small numbers (every Mersenne prime, the first five perfect numbers) are there. Other sats lost for other reasons (for example the two overwritten duplicate-coinbase outputs of 2010) are not flagged.

## Considered and left out

We looked at many other famous number classes. A label only earns a place if some sat you could actually own carries it, if we can compute it ourselves from the definition, and if it is not so common that it means nothing.

**No member among the sat numbers at all.**
- *Odd perfect numbers:* none exist below 10¹⁵⁰⁰ (Ochem and Rao, 2012), and sat numbers stop near 2.1 × 10¹⁵. (This is also why our perfect-number list is complete.)
- *Wall–Sun–Sun primes:* none below 2⁶⁴ (PrimeGrid, finished 2022).
- *Odd weird numbers:* none below 10²¹.
- *Repunit primes* beyond 11, *Belphegor primes* beyond 16,661, Mersenne and twin-prime *records*, and *Ta(5)* onwards are all far larger than the last sat.

**Every known member is inside the genesis block, so no one can own one.** Every known member of these classes is below 5,000,000,000, and a label that can only ever say "unspendable" adds nothing to a lookup.
- *Fermat primes* (3, 5, 17, 257, 65,537): the next Fermat number is about 1.8 × 10¹⁹, past the last sat.
- *Wieferich primes* (1,093 and 3,511): searches found no other below 2⁶⁴, far past the last sat.
- *Wilson primes* (5, 13, 563) and *Wolstenholme primes* (16,843 and 2,124,679): searches reach 2 × 10¹³ and 10¹¹, short of the last sat. A spendable one isn't ruled out, but none is known, and the rest of the search is beyond what we can run.
- *Kaprekar's constant* (6,174) and *sublime numbers*: 6,174 is a single number. Only two sublime numbers are known, 12 and one with 76 digits.

**Too common to mean anything.** Twin primes, Sophie Germain primes, cousin and sexy primes, and emirps each cover trillions of sats, about as many as the primes themselves. Harshad, happy, Smith and lucky numbers are denser still. These may come back as extra "common" labels if `prime` stays on.

**One sat is easy to check, but we can't list them all ourselves.** Carmichael numbers (composites that pass Fermat's primality test for every base they share no factor with), amicable numbers (pairs like 220 and 284, where each is the sum of the other's proper divisors) and Keith numbers (like 14: 1, 4, 5, 9, 14) can each be checked for a single sat in well under a second. The hard part is the complete list. The rare labels above each come with every member below the last sat, which is what the notable list and the search inside a wallet's sat ranges are built on, and our tests rebuild every list a second way to check it. Listing every Carmichael number (more than 100,000 below the last sat) or every amicable pair takes a long specialised search; we would have to import someone else's published table rather than compute and check it ourselves. Keith numbers need a search that grows tenfold with each extra digit; faster methods exist, but for a base-10 curiosity they aren't worth the work. Carmichael numbers may come back as a label that shows only when you look up a single sat.

**Not settled.** No number has been proven to be a Lychrel number (one that never becomes a palindrome by reversing and adding; 196 is the famous candidate). A label for "probably" is not one we want to give.

## Known limits

- A label is a fact about a number, not a market price. Whether anyone values a `fibonacci` sat more than its neighbour is exactly what this experiment is asking.
- Looking up an inscription's sat needs our own ord index with sat numbers indexed; when ord can't say, the lookup says so rather than guessing.
- Which pattern labels are shown, and whether `prime` is shown at all, are settings and may change while the experiment runs.