Block #310,117

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/13/2013, 10:30:44 PM · Difficulty 9.9951 · 6,485,347 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

e5a080222016f827ec71c4d344fe2f48ec25dd2f2e05b992565ca0b9f0fc4323

View on Chainz ↗

Height

#310,117

Difficulty

9.995058

Transactions

Size

1.14 KB

Version

Bits

09febc25

Nonce

50,251

Timestamp

12/13/2013, 10:30:44 PM

Confirmations

6,485,347

Mined by

⛏️ eaRAYtDvcGsMH2GY5bD4Hc2rArhMdVuAcA7m7

Merkle Root

da0bd48a26747d086aa2d436110104f71c36092a7a36659c0e6be8b72b46e774

Transactions (1)

Coinbaseda0bd48a26747d…6be8b72b46e774

1 in → 30 out9.9700 XPM1.06 KB

AYtDvcGs…VuAcA7m7 Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz AcM3ext6…Hwtj9Qp3 AMbCuLHZ…WSoStwkc

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.

4.526 × 10⁹⁰(91-digit number)

45261314569296570050…28959319692602998399

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

4.526 × 10⁹⁰(91-digit number)

45261314569296570050…28959319692602998399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

4.526 × 10⁹⁰(91-digit number)

45261314569296570050…28959319692602998401

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

9.052 × 10⁹⁰(91-digit number)

90522629138593140100…57918639385205996799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

9.052 × 10⁹⁰(91-digit number)

90522629138593140100…57918639385205996801

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

1.810 × 10⁹¹(92-digit number)

18104525827718628020…15837278770411993599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.810 × 10⁹¹(92-digit number)

18104525827718628020…15837278770411993601

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

3.620 × 10⁹¹(92-digit number)

36209051655437256040…31674557540823987199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

3.620 × 10⁹¹(92-digit number)

36209051655437256040…31674557540823987201

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

7.241 × 10⁹¹(92-digit number)

72418103310874512080…63349115081647974399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

7.241 × 10⁹¹(92-digit number)

72418103310874512080…63349115081647974401

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