Block #265,610

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/19/2013, 5:41:36 PM · Difficulty 9.9621 · 6,537,667 confirmations

TWN

Bi-Twin Chain

Interleaved pairs of primes that differ by 2, forming twin prime pairs at each level.

Block Header

Block Hash

adaaf39b70df85fffd6cc1405d44a2b9f594fdcb41383bb92ab446d28c8ba769

View on Chainz ↗

Height

#265,610

Difficulty

9.962140

Transactions

Size

1.81 KB

Version

Bits

09f64ec8

Nonce

55,312

Timestamp

11/19/2013, 5:41:36 PM

Confirmations

6,537,667

Mined by

⛏️ aR>AP5WHbwXb9xSBo3xgmawzr4QC843rw7Y9N

Merkle Root

eaace1a90f4232af0d2aad9846a106596fe6a835f549f224e5d0ce1e941078fd

Transactions (1)

Coinbaseeaace1a90f4232…d0ce1e941078fd

1 in → 50 out10.0400 XPM1.72 KB

AP5WHbwX…43rw7Y9N AKXeSXzu…yeuNjvhB APZy8y6E…QZaGQjfp ANHbaxZp…XjnHCJJs AVzhvfTX…ZzNJYq61

Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) — it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

2.931 × 10⁸⁶(87-digit number)

29319268004752622977…90975992384187756159

Discovered Prime Numbers

Lower: 2^k × origin − 1 | Upper: 2^k × origin + 1

These are the actual prime numbers discovered by this block, computed using the verified Primecoin formula. Each number has been independently confirmed to pass the Fermat primality test. Use the FactorDB links to verify any number independently.

Level 0 — Twin Prime Pair (origin ± 1)

origin − 1

2.931 × 10⁸⁶(87-digit number)

29319268004752622977…90975992384187756159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.931 × 10⁸⁶(87-digit number)

29319268004752622977…90975992384187756161

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: origin + 1 − origin − 1 = 2 (twin primes ✓)

Level 1 — Twin Prime Pair (2^1 × origin ± 1)

2^1 × origin − 1

5.863 × 10⁸⁶(87-digit number)

58638536009505245955…81951984768375512319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.863 × 10⁸⁶(87-digit number)

58638536009505245955…81951984768375512321

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^1 × origin + 1 − 2^1 × origin − 1 = 2 (twin primes ✓)

Level 2 — Twin Prime Pair (2^2 × origin ± 1)

2^2 × origin − 1

1.172 × 10⁸⁷(88-digit number)

11727707201901049191…63903969536751024639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.172 × 10⁸⁷(88-digit number)

11727707201901049191…63903969536751024641

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^2 × origin + 1 − 2^2 × origin − 1 = 2 (twin primes ✓)

Level 3 — Twin Prime Pair (2^3 × origin ± 1)

2^3 × origin − 1

2.345 × 10⁸⁷(88-digit number)

23455414403802098382…27807939073502049279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.345 × 10⁸⁷(88-digit number)

23455414403802098382…27807939073502049281

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

4.691 × 10⁸⁷(88-digit number)

46910828807604196764…55615878147004098559

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin — the large number shown above — anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

★☆☆☆☆

Rarity

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

How Primecoin's Proof-of-Work Constructs These Primes

Primecoin stores a value called the prime chain origin in each block. The miner found a large integer such that when divided by a primorial (the product of small primes: 2 × 3 × 5 × 7 × …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial — that divisibility is part of the proof.

Prime Chain Origin = First Prime × Primorial (2·3·5·7·11·…)

Source: Primecoin prime.cpp — CheckPrimeProofOfWork()

This is why the origin has many small prime factors — those factors are the primorial divisor. The chain then extends from the first prime using the TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer