Block #315,248

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/16/2013, 10:10:43 AM · Difficulty 10.0914 · 6,484,020 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

2e91c1d92e24a67e40b31b0d616010282396e36ce3fc6332ae479f174ee45529

View on Chainz ↗

Height

#315,248

Difficulty

10.091442

Transactions

Size

1.74 KB

Version

Bits

0a1768c1

Nonce

16,632

Timestamp

12/16/2013, 10:10:43 AM

Confirmations

6,484,020

Mined by

⛏️ paRAd9pSzHtZ9ebPysWxo4VY8quTmcpJ3y2zz

Merkle Root

981945a16a11019f89c2fde7abb0f23b7e86ff3209bf4b51fd31d2461ab33f06

Transactions (4)

Coinbase0e0a900f5e786e…4e7959fb3d8b20

1 in → 28 out9.7800 XPM1015 B

Ad9pSzHt…cpJ3y2zz Aac9Hjdg…P4ktypc3 AHZUHC8g…t3rBwmSA AdU5vssM…hMrSqTaA AcJjE2Eg…ybyTQuYb

48dedd9cf57f46…144117ffe41a00

1 in → 2 out435.4771 XPM226 B

AUVfGWct…TN5b3Pm4 AcWN5Hkz…QNhFwomF

303e8a51bc3650…7f317393b49f7a

1 in → 2 out4.7300 XPM227 B

AVvyhacf…VgfUiyaR AemByUL3…yYbXSy8D

396315f9215ebe…89cdc7217e294a

1 in → 2 out7.8582 XPM226 B

AUQGr2M5…gPMYycuE ARrNbZTN…xcM7Gquo

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.

2.976 × 10⁹⁷(98-digit number)

29763013685193604851…91348886813373515599

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

2.976 × 10⁹⁷(98-digit number)

29763013685193604851…91348886813373515599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.976 × 10⁹⁷(98-digit number)

29763013685193604851…91348886813373515601

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

5.952 × 10⁹⁷(98-digit number)

59526027370387209702…82697773626747031199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

5.952 × 10⁹⁷(98-digit number)

59526027370387209702…82697773626747031201

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

1.190 × 10⁹⁸(99-digit number)

11905205474077441940…65395547253494062399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.190 × 10⁹⁸(99-digit number)

11905205474077441940…65395547253494062401

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

2.381 × 10⁹⁸(99-digit number)

23810410948154883880…30791094506988124799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.381 × 10⁹⁸(99-digit number)

23810410948154883880…30791094506988124801

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

4.762 × 10⁹⁸(99-digit number)

47620821896309767761…61582189013976249599

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

4.762 × 10⁹⁸(99-digit number)

47620821896309767761…61582189013976249601

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