Block #315,865

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/16/2013, 6:09:47 PM · Difficulty 10.1163 · 6,493,685 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

65e9b88fdf992319d71bae3d8f054c9f4ef97667e2378b812c9478c276103c03

View on Chainz ↗

Height

#315,865

Difficulty

10.116300

Transactions

Size

1.01 KB

Version

Bits

0a1dc5d4

Nonce

19,654

Timestamp

12/16/2013, 6:09:47 PM

Confirmations

6,493,685

Mined by

⛏️ aRAAQwRN8bEjE7sawFBq7N38V9niAV8PsTcY9

Merkle Root

06d04994e289ae39171ddff594b9777f2a90fb31d48d4d0d7567ba3474af5ee1

Transactions (1)

Coinbase06d04994e289ae…67ba3474af5ee1

1 in → 26 out9.7600 XPM947 B

AQwRN8bE…V8PsTcY9 AYtDvcGs…VuAcA7m7 Aac9Hjdg…P4ktypc3 AVdK3UJg…LiLMfWGV AVGz7MWQ…EGc1BNJT

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.

5.700 × 10⁹¹(92-digit number)

57007079002142348086…65975423645976267399

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

5.700 × 10⁹¹(92-digit number)

57007079002142348086…65975423645976267399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

5.700 × 10⁹¹(92-digit number)

57007079002142348086…65975423645976267401

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

1.140 × 10⁹²(93-digit number)

11401415800428469617…31950847291952534799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.140 × 10⁹²(93-digit number)

11401415800428469617…31950847291952534801

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

2.280 × 10⁹²(93-digit number)

22802831600856939234…63901694583905069599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

2.280 × 10⁹²(93-digit number)

22802831600856939234…63901694583905069601

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

4.560 × 10⁹²(93-digit number)

45605663201713878469…27803389167810139199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

4.560 × 10⁹²(93-digit number)

45605663201713878469…27803389167810139201

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

9.121 × 10⁹²(93-digit number)

91211326403427756938…55606778335620278399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

9.121 × 10⁹²(93-digit number)

91211326403427756938…55606778335620278401

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 #315,865

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