Block #310,763

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/14/2013, 5:41:42 AM · Difficulty 9.9953 · 6,491,722 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

2b4fe6c587e33ebf026e5b94f9ddc3eaebe25c92822b4d7f91425199bec00fd3

View on Chainz ↗

Height

#310,763

Difficulty

9.995284

Transactions

Size

2.17 KB

Version

Bits

09fecaed

Nonce

62,996

Timestamp

12/14/2013, 5:41:42 AM

Confirmations

6,491,722

Mined by

⛏️ aRAac9Hjdgnw4T4L2csqLecJsmZjP4ktypc3

Merkle Root

78ee1b84a86a6d4d54ddf197a669d8e0b1fa50d55188e3af7c10cbf5fb7aeed2

Transactions (4)

Coinbase0cd42900df5ad4…3ad13af55790b3

1 in → 29 out9.9700 XPM1.02 KB

Aac9Hjdg…P4ktypc3 Ad9pSzHt…cpJ3y2zz ASWX42VM…XypQABkY AbtgNzNb…9Atvu81w ASy3AE9X…ghJMwaMJ

bd03d056b9378d…7f36ebf56cf60f

2 in → 1 out2.0000 XPM339 B

ASAnf3SC…di1Mt2CE

23243e993c5b44…ccdbc0eb7800ef

2 in → 2 out15.6199 XPM374 B

Aa1AvAie…Yv4xhXZC AabR8q8G…uJNR8DpK

082668a9be2649…344530cdac4cf7

2 in → 2 out13.9800 XPM373 B

AQEtHhbU…JTXgnneZ Aa1AvAie…Yv4xhXZC

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.

9.571 × 10⁸⁷(88-digit number)

95712596438685829641…86466952975929235199

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

9.571 × 10⁸⁷(88-digit number)

95712596438685829641…86466952975929235199

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

9.571 × 10⁸⁷(88-digit number)

95712596438685829641…86466952975929235201

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

1.914 × 10⁸⁸(89-digit number)

19142519287737165928…72933905951858470399

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.914 × 10⁸⁸(89-digit number)

19142519287737165928…72933905951858470401

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

3.828 × 10⁸⁸(89-digit number)

38285038575474331856…45867811903716940799

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.828 × 10⁸⁸(89-digit number)

38285038575474331856…45867811903716940801

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

7.657 × 10⁸⁸(89-digit number)

76570077150948663713…91735623807433881599

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

7.657 × 10⁸⁸(89-digit number)

76570077150948663713…91735623807433881601

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

1.531 × 10⁸⁹(90-digit number)

15314015430189732742…83471247614867763199

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

1.531 × 10⁸⁹(90-digit number)

15314015430189732742…83471247614867763201

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