Block #291,505

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 12/3/2013, 5:36:40 AM · Difficulty 9.9899 · 6,508,018 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

39e8fbe809ff7dff17a63e4c27d3f0b177c91015cd0c82bdd20fecc43423de47

View on Chainz ↗

Height

#291,505

Difficulty

9.989875

Transactions

Size

528 B

Version

Bits

09fd6874

Nonce

316,118

Timestamp

12/3/2013, 5:36:40 AM

Confirmations

6,508,018

Mined by

⛏️ raRmANuKx9Kemb4q2j6b7vjmfuKrAuwm7nrBRq

Merkle Root

e931faa50187e6f66917ff39b5f51b6d499cdc2cd9654e5c57c3489e55f659f0

Transactions (1)

Coinbasee931faa50187e6…c3489e55f659f0

1 in → 11 out10.0100 XPM437 B

ANuKx9Ke…wm7nrBRq AHUtCSic…9RMQggzJ AMi72qfB…rp5J81ry AJzWx2VD…j4ZSMit5 ARXSrG7z…CRW69WR4

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.

5.737 × 10⁹⁷(98-digit number)

57372351292762403240…32578379374730408959

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

5.737 × 10⁹⁷(98-digit number)

57372351292762403240…32578379374730408959

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.737 × 10⁹⁷(98-digit number)

57372351292762403240…32578379374730408961

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

1.147 × 10⁹⁸(99-digit number)

11474470258552480648…65156758749460817919

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.147 × 10⁹⁸(99-digit number)

11474470258552480648…65156758749460817921

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

2.294 × 10⁹⁸(99-digit number)

22948940517104961296…30313517498921635839

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.294 × 10⁹⁸(99-digit number)

22948940517104961296…30313517498921635841

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

4.589 × 10⁹⁸(99-digit number)

45897881034209922592…60627034997843271679

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.589 × 10⁹⁸(99-digit number)

45897881034209922592…60627034997843271681

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

9.179 × 10⁹⁸(99-digit number)

91795762068419845184…21254069995686543359

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