Block #311,319

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 12:39:48 PM · Difficulty 9.9954 · 6,491,190 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

0077533cb6215382230f78b4a1db1bfc0e68b682b7c79b32fe91ff49e835c9e3

View on Chainz ↗

Height

#311,319

Difficulty

9.995429

Transactions

Size

1.18 KB

Version

Bits

09fed468

Nonce

58,017

Timestamp

12/14/2013, 12:39:48 PM

Confirmations

6,491,190

Mined by

⛏️ aRQuAac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

a9cc79e952a092fd1375a9bde733ef4bddb86a1c1e94dc3607f2a4784e88f96b

Transactions (1)

Coinbasea9cc79e952a092…f2a4784e88f96b

1 in → 31 out9.9700 XPM1.09 KB

Aac9Hjdg…P4ktypc3 AeNZE3ry…LeNEbiVb AZ2pPuKg…95SN2Yr7 ASWX42VM…XypQABkY AcjhLWWR…QEkog6Bq

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.

1.616 × 10⁹³(94-digit number)

16167582258118063909…06393139091773828049

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

1.616 × 10⁹³(94-digit number)

16167582258118063909…06393139091773828049

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.616 × 10⁹³(94-digit number)

16167582258118063909…06393139091773828051

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

3.233 × 10⁹³(94-digit number)

32335164516236127818…12786278183547656099

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

3.233 × 10⁹³(94-digit number)

32335164516236127818…12786278183547656101

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

6.467 × 10⁹³(94-digit number)

64670329032472255636…25572556367095312199

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

6.467 × 10⁹³(94-digit number)

64670329032472255636…25572556367095312201

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

1.293 × 10⁹⁴(95-digit number)

12934065806494451127…51145112734190624399

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.293 × 10⁹⁴(95-digit number)

12934065806494451127…51145112734190624401

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

2.586 × 10⁹⁴(95-digit number)

25868131612988902254…02290225468381248799

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.586 × 10⁹⁴(95-digit number)

25868131612988902254…02290225468381248801

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