Block #264,066

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 11/18/2013, 9:05:51 AM · Difficulty 9.9650 · 6,539,703 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e76865af5da2e6d76fbc471dea50e5faaa3c26d0d2ddfd144fa9578034f6c970

View on Chainz ↗

Height

#264,066

Difficulty

9.965046

Transactions

Size

1.74 KB

Version

Bits

09f70d46

Nonce

12,418

Timestamp

11/18/2013, 9:05:51 AM

Confirmations

6,539,703

Mined by

⛏️ P2SHAdGRRh1eP4JFvQMqy7y2pLsryDQmctLb24

Merkle Root

57ea6bccff7d7e8576f9b9dd9528186c28f4f521f45305e2eca23bcf4956117e

Transactions (4)

Coinbase257b2a88840759…124b74807f3e25

1 in → 2 out10.1000 XPM142 B

AdGRRh1e…QmctLb24 AKNbfPoc…jfkpwAyL

86157a87676aef…fea51974c57995

6 in → 8 out22.2564 XPM1.08 KB

AXWkTuvV…rjTvyqaL AWn9rWeZ…sxk6DT9k AYjwrrNa…p6r7yfsV AHb9D17o…rg9iXrZC AeKNrjNj…1NEq95Ta

228f4c35a46780…c82f42165020b7

1 in → 2 out8.0077 XPM226 B

Ac1Gzkq7…19xjeTrj AMw1gR1M…NmjnMW9P

0558c37f724de8…679c727fbe3ad3

1 in → 2 out2.4409 XPM225 B

AJnakggQ…cyZiViWJ AMBAxdJX…SfVSy35j

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.

1.366 × 10⁹⁵(96-digit number)

13669080797457257629…59547228653734217119

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

1.366 × 10⁹⁵(96-digit number)

13669080797457257629…59547228653734217119

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.366 × 10⁹⁵(96-digit number)

13669080797457257629…59547228653734217121

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

2.733 × 10⁹⁵(96-digit number)

27338161594914515259…19094457307468434239

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.733 × 10⁹⁵(96-digit number)

27338161594914515259…19094457307468434241

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

5.467 × 10⁹⁵(96-digit number)

54676323189829030518…38188914614936868479

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.467 × 10⁹⁵(96-digit number)

54676323189829030518…38188914614936868481

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

1.093 × 10⁹⁶(97-digit number)

10935264637965806103…76377829229873736959

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.093 × 10⁹⁶(97-digit number)

10935264637965806103…76377829229873736961

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

2.187 × 10⁹⁶(97-digit number)

21870529275931612207…52755658459747473919

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