Block #404,787

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 2/15/2014, 2:41:50 AM · Difficulty 10.4325 · 6,389,499 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

2d41aa57d93d51e080df192c2751b8e8915ebd0c79fa4be534dccc0518f36d56

View on Chainz ↗

Height

#404,787

Difficulty

10.432543

Transactions

Size

1.94 KB

Version

Bits

0a6ebb2b

Nonce

192,571

Timestamp

2/15/2014, 2:41:50 AM

Confirmations

6,389,499

Merkle Root

dac1a47a9b54d23d8e7635fbcdb9c4e3b0a6ae8a09701642e1a6427efa1f1580

Transactions (4)

Coinbaseadccaf45014613…ec1677adc79622

1 in → 23 out9.1700 XPM845 B

AQvZBsUo…hppgRZTJ AZV4TwxQ…8JnQqSoH Aac9Hjdg…P4ktypc3 AGKhfQo2…h7fBXLX6 ALqxX1WA…hsKjbnXn

b512146b13972f…36f2a1ec5c87f6

3 in → 1 out52.8141 XPM488 B

ANzpeU12…xvbGzWYU

c9394a5d186982…c49e2d3d943bac

1 in → 1 out2.9919 XPM192 B

AWvjJGfi…LAHUcBRf

0fa3bada40b4af…1f47d6925ca705

2 in → 2 out3.8261 XPM373 B

AW1o6TBj…KP6hSLnx ARdBiZtb…nNEaM5Sk

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.

7.021 × 10⁹⁸(99-digit number)

70210983398891630941…44307630863301456199

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

7.021 × 10⁹⁸(99-digit number)

70210983398891630941…44307630863301456199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

7.021 × 10⁹⁸(99-digit number)

70210983398891630941…44307630863301456201

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

1.404 × 10⁹⁹(100-digit number)

14042196679778326188…88615261726602912399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.404 × 10⁹⁹(100-digit number)

14042196679778326188…88615261726602912401

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

2.808 × 10⁹⁹(100-digit number)

28084393359556652376…77230523453205824799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.808 × 10⁹⁹(100-digit number)

28084393359556652376…77230523453205824801

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

5.616 × 10⁹⁹(100-digit number)

56168786719113304752…54461046906411649599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

5.616 × 10⁹⁹(100-digit number)

56168786719113304752…54461046906411649601

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

1.123 × 10¹⁰⁰(101-digit number)

11233757343822660950…08922093812823299199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.123 × 10¹⁰⁰(101-digit number)

11233757343822660950…08922093812823299201

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^4 × origin + 1 − 2^4 × origin − 1 = 2 (twin primes ✓)

What this block proved

The miner who found this block proved the existence of 10 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

UncommonChain length 10

Roughly 1 in 100 blocks. Solid but expected in a healthy network.

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