Block #311,097

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 9:48:22 AM · Difficulty 9.9954 · 6,501,649 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

9204eade1c1ca1f54c3dfcd766762ec77eaefcc1e29e6eb3067aaf5f25e424f4

View on Chainz ↗

Height

#311,097

Difficulty

9.995375

Transactions

Size

527 B

Version

Bits

09fed0ec

Nonce

156,857

Timestamp

12/14/2013, 9:48:22 AM

Confirmations

6,501,649

Mined by

⛏️ 9aRAbopjr7ZNGrUjb1L7deX9XNozHzLTB81mz

Merkle Root

7dd8c15cb35dd377b182a1e43f0e9aac80fc594183b12532c0a1fd2613a7a56b

Transactions (2)

Coinbase561fd483dfa6de…3f41de6a22cfda

1 in → 1 out9.9900 XPM97 B

Abopjr7Z…zLTB81mz

f85d1404ddb27c…fa0def50f0e48b

2 in → 1 out6.0944 XPM341 B

AcmNSgfj…YFvs8RwQ

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.

1.442 × 10⁹¹(92-digit number)

14429819515364888942…70578405009945868559

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

1.442 × 10⁹¹(92-digit number)

14429819515364888942…70578405009945868559

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

1.442 × 10⁹¹(92-digit number)

14429819515364888942…70578405009945868561

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

2.885 × 10⁹¹(92-digit number)

28859639030729777885…41156810019891737119

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

2.885 × 10⁹¹(92-digit number)

28859639030729777885…41156810019891737121

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

5.771 × 10⁹¹(92-digit number)

57719278061459555770…82313620039783474239

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

5.771 × 10⁹¹(92-digit number)

57719278061459555770…82313620039783474241

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

1.154 × 10⁹²(93-digit number)

11543855612291911154…64627240079566948479

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.154 × 10⁹²(93-digit number)

11543855612291911154…64627240079566948481

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

2.308 × 10⁹²(93-digit number)

23087711224583822308…29254480159133896959

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

2.308 × 10⁹²(93-digit number)

23087711224583822308…29254480159133896961

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 #311,097

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