Block #313,097

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/15/2013, 7:46:30 AM · Difficulty 9.9960 · 6,495,841 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

8e0c079303314c3d2261c3a417607be53ff19f16ddf62a9065fbc06ba294b7f2

View on Chainz ↗

Height

#313,097

Difficulty

9.996021

Transactions

Size

2.69 KB

Version

Bits

09fefb34

Nonce

652

Timestamp

12/15/2013, 7:46:30 AM

Confirmations

6,495,841

Mined by

⛏️ aR^#_Aac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

1929207559c5bd915ec58d39cf71545f7e30ea7c047941de2b6faf20931f9b0c

Transactions (4)

Coinbase5863de98ac2652…f133a07bb5e1b4

1 in → 30 out9.9700 XPM1.06 KB

Aac9Hjdg…P4ktypc3 ASWX42VM…XypQABkY AZ2pPuKg…95SN2Yr7 AbtgNzNb…9Atvu81w Ac6mGvmQ…XqWmPHjR

905733d1ca67eb…917ff755cab658

3 in → 2 out14.0120 XPM522 B

AJKE93bL…PYz14RpR ANfAxW1R…c1655jpu

367c98d00ea3f6…3b38a4d04a0b85

4 in → 11 out40.1900 XPM839 B

ATMR5B9b…ALqiXPCF AZ4xYNkB…N2CsF6dq AVMCwCki…wC16T1HN APFo1nYS…GGFw32mR AcADmAoe…8iNnoMzB

dd674761646152…1df68f7c9788e9

1 in → 2 out5.7938 XPM226 B

APUAzVy5…zN87gdfa ALkuF5Je…KBUB19wB

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.

2.359 × 10⁹²(93-digit number)

23591131110185820576…97873574661112032519

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

2.359 × 10⁹²(93-digit number)

23591131110185820576…97873574661112032519

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.359 × 10⁹²(93-digit number)

23591131110185820576…97873574661112032521

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

4.718 × 10⁹²(93-digit number)

47182262220371641153…95747149322224065039

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.718 × 10⁹²(93-digit number)

47182262220371641153…95747149322224065041

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

9.436 × 10⁹²(93-digit number)

94364524440743282307…91494298644448130079

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.436 × 10⁹²(93-digit number)

94364524440743282307…91494298644448130081

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

1.887 × 10⁹³(94-digit number)

18872904888148656461…82988597288896260159

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.887 × 10⁹³(94-digit number)

18872904888148656461…82988597288896260161

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

3.774 × 10⁹³(94-digit number)

37745809776297312922…65977194577792520319

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.774 × 10⁹³(94-digit number)

37745809776297312922…65977194577792520321

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 #313,097

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