Block #291,027

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/2/2013, 11:43:37 PM · Difficulty 9.9896 · 6,519,553 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

6e7408d2f11f9030f3c6597799043818b1f1c1c92391d82e550d72203c1137f9

View on Chainz ↗

Height

#291,027

Difficulty

9.989604

Transactions

Size

1.15 KB

Version

Bits

09fd56b3

Nonce

85,918

Timestamp

12/2/2013, 11:43:37 PM

Confirmations

6,519,553

Mined by

⛏️ paRAZninDSu1Gy3NdWkaTGfLDa2i9FvgDMwC4

Merkle Root

e16d53911c8c99197dd701343496b8ac6f4112b2c28400db7cbf65e5bc9fdce5

Transactions (1)

Coinbasee16d53911c8c99…bf65e5bc9fdce5

1 in → 30 out9.9900 XPM1.06 KB

AZninDSu…FvgDMwC4 AcC2zSod…rtWatH79 AdyKWr2A…7PSnFqeJ AU3tFYHS…gK4SzeuL AHLsU39G…fiNcno66

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.

4.260 × 10⁹⁶(97-digit number)

42602085707497970878…25142696132508449279

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

4.260 × 10⁹⁶(97-digit number)

42602085707497970878…25142696132508449279

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.260 × 10⁹⁶(97-digit number)

42602085707497970878…25142696132508449281

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

8.520 × 10⁹⁶(97-digit number)

85204171414995941756…50285392265016898559

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

8.520 × 10⁹⁶(97-digit number)

85204171414995941756…50285392265016898561

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

1.704 × 10⁹⁷(98-digit number)

17040834282999188351…00570784530033797119

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.704 × 10⁹⁷(98-digit number)

17040834282999188351…00570784530033797121

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

3.408 × 10⁹⁷(98-digit number)

34081668565998376702…01141569060067594239

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.408 × 10⁹⁷(98-digit number)

34081668565998376702…01141569060067594241

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

6.816 × 10⁹⁷(98-digit number)

68163337131996753404…02283138120135188479

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

6.816 × 10⁹⁷(98-digit number)

68163337131996753404…02283138120135188481

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 #291,027

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