Block #317,395

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/17/2013, 4:16:19 PM · Difficulty 10.1517 · 6,477,256 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

0051dddf4e62a3697002da734237484e21b517a4c1d79dcc7c89506f66250cf1

View on Chainz ↗

Height

#317,395

Difficulty

10.151669

Transactions

Size

1.85 KB

Version

Bits

0a26d3ca

Nonce

74,587

Timestamp

12/17/2013, 4:16:19 PM

Confirmations

6,477,256

Mined by

⛏️ aRx0AJzWx2VDShfKP9pKK3jZqTbn4sj4ZSMit5

Merkle Root

74359408a6861085f950e7477f1e8913f689ead4ab129e38f3b5a78024cf923a

Transactions (2)

Coinbase10beb41ac0d443…7130cbb7b5208c

1 in → 23 out9.6900 XPM845 B

AJzWx2VD…j4ZSMit5 AJYusoxy…LtcN2Chh Aac9Hjdg…P4ktypc3 AYtDvcGs…VuAcA7m7 AQyr8Lw3…hs4nt3Pn

f99673556f7e64…03ef34471435dd

5 in → 8 out21.3304 XPM954 B

AJaQENaD…wRN4nbo6 AeKNrjNj…1NEq95Ta AdrVFPwD…WbnMHNBa AM7Tr7C5…DnAdRwkh AZoyjqWo…cTWTRwva

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.

3.375 × 10⁹⁸(99-digit number)

33751025950869918775…16715397663917944159

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

3.375 × 10⁹⁸(99-digit number)

33751025950869918775…16715397663917944159

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.375 × 10⁹⁸(99-digit number)

33751025950869918775…16715397663917944161

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

6.750 × 10⁹⁸(99-digit number)

67502051901739837551…33430795327835888319

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

6.750 × 10⁹⁸(99-digit number)

67502051901739837551…33430795327835888321

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

1.350 × 10⁹⁹(100-digit number)

13500410380347967510…66861590655671776639

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.350 × 10⁹⁹(100-digit number)

13500410380347967510…66861590655671776641

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

2.700 × 10⁹⁹(100-digit number)

27000820760695935020…33723181311343553279

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.700 × 10⁹⁹(100-digit number)

27000820760695935020…33723181311343553281

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

5.400 × 10⁹⁹(100-digit number)

54001641521391870041…67446362622687106559

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

5.400 × 10⁹⁹(100-digit number)

54001641521391870041…67446362622687106561

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