Block #266,404

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 11/20/2013, 10:17:52 AM · Difficulty 9.9606 · 6,543,287 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

ad5d56e6949767bda259012d008d1b6012ec57311b6c97f22e4c05f7b5973cfe

View on Chainz ↗

Height

#266,404

Difficulty

9.960602

Transactions

Size

4.66 KB

Version

Bits

09f5ea05

Nonce

2,160

Timestamp

11/20/2013, 10:17:52 AM

Confirmations

6,543,287

Mined by

⛏️ P2SHAdGRRh1eP4JFvQMqy7y2pLsryDQmctLb24

Merkle Root

4c0653dd4fa95c772323748f35e199c22b850a5e118c12ab7fae4b1644403be4

Transactions (4)

Coinbasecd9fb9823d6d69…8f1dcda4b13d09

1 in → 2 out10.1200 XPM142 B

AdGRRh1e…QmctLb24 AKNbfPoc…jfkpwAyL

00566bb3c504f8…d7bb00d0a9be9f

5 in → 2 out308.4439 XPM818 B

AQvDgmA8…2a8vAckz ASF31wYi…42B7qC7u

d1200bef466c84…e63aad483cf1fc

2 in → 1 out20.0900 XPM271 B

AMsGxSh9…ka9LXJoe

3041813e7899f5…808f589c16725d

23 in → 1 out3.1436 XPM3.37 KB

ANv7Zfs9…maQ7fuTk

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.

2.675 × 10⁹⁹(100-digit number)

26750503818497294321…95038472085718127299

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

2.675 × 10⁹⁹(100-digit number)

26750503818497294321…95038472085718127299

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.675 × 10⁹⁹(100-digit number)

26750503818497294321…95038472085718127301

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

5.350 × 10⁹⁹(100-digit number)

53501007636994588642…90076944171436254599

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.350 × 10⁹⁹(100-digit number)

53501007636994588642…90076944171436254601

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

1.070 × 10¹⁰⁰(101-digit number)

10700201527398917728…80153888342872509199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.070 × 10¹⁰⁰(101-digit number)

10700201527398917728…80153888342872509201

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

2.140 × 10¹⁰⁰(101-digit number)

21400403054797835456…60307776685745018399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.140 × 10¹⁰⁰(101-digit number)

21400403054797835456…60307776685745018401

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

4.280 × 10¹⁰⁰(101-digit number)

42800806109595670913…20615553371490036799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.280 × 10¹⁰⁰(101-digit number)

42800806109595670913…20615553371490036801

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

Block #266,404

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