Block #312,862

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/15/2013, 5:24:21 AM · Difficulty 9.9959 · 6,479,912 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

7d3c342d99b95381edc973bc702c1943387b37b55cc0f0fabd81e21a53d126a3

View on Chainz ↗

Height

#312,862

Difficulty

9.995940

Transactions

Size

1.11 KB

Version

Bits

09fef5e6

Nonce

111,984

Timestamp

12/15/2013, 5:24:21 AM

Confirmations

6,479,912

Merkle Root

c7b6d07a383c368939229db9644f95337b3ee39ee6d43c1128d894e0bab14d99

Transactions (1)

Coinbasec7b6d07a383c36…d894e0bab14d99

1 in → 29 out9.9700 XPM1.02 KB

Aac9Hjdg…P4ktypc3 AKzkYQaQ…zR4FspJZ AP9eJ2CZ…eYfb95GN AbDHrhkn…fbZN1SRr APeshfYV…3PKcKMKH

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.

2.385 × 10⁸⁸(89-digit number)

23859559838474430951…55352961045243163399

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

2.385 × 10⁸⁸(89-digit number)

23859559838474430951…55352961045243163399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.385 × 10⁸⁸(89-digit number)

23859559838474430951…55352961045243163401

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

4.771 × 10⁸⁸(89-digit number)

47719119676948861902…10705922090486326799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.771 × 10⁸⁸(89-digit number)

47719119676948861902…10705922090486326801

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

9.543 × 10⁸⁸(89-digit number)

95438239353897723805…21411844180972653599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

9.543 × 10⁸⁸(89-digit number)

95438239353897723805…21411844180972653601

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

1.908 × 10⁸⁹(90-digit number)

19087647870779544761…42823688361945307199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.908 × 10⁸⁹(90-digit number)

19087647870779544761…42823688361945307201

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

3.817 × 10⁸⁹(90-digit number)

38175295741559089522…85647376723890614399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.817 × 10⁸⁹(90-digit number)

38175295741559089522…85647376723890614401

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