Block #331,121

TWNLength 10★★☆☆☆

Bi-Twin Chain · Discovered 12/27/2013, 4:01:04 AM · Difficulty 10.1657 · 6,472,256 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

d94c70d14a249b90ba475a5a101978b85f01d3442e31ba0de89a39706e13fd21

View on Chainz ↗

Height

#331,121

Difficulty

10.165685

Transactions

Size

9.67 KB

Version

Bits

0a2a6a5b

Nonce

43,471

Timestamp

12/27/2013, 4:01:04 AM

Confirmations

6,472,256

Mined by

⛏️ P2SHAeKNrjNjtSncLJjUrDnQbrfzMS1NEq95Ta

Merkle Root

054b12c9dd2d7de716d859530ef222b023b5213ce9cd29e5e213f90999e6c0ae

Transactions (4)

Coinbase13dea2a03cd52e…e902675957b952

1 in → 1 out9.7800 XPM110 B

AeKNrjNj…1NEq95Ta

86d89bd39de457…95b136ad8e9f6c

34 in → 2 out34.0216 XPM4.99 KB

APmPZi1m…chVMQ2aJ AbBH5GvN…aMiNWtHu

4f3fc8c122fd2e…73ebe69aa9e56d

29 in → 2 out10.0100 XPM4.27 KB

AVomM6tZ…bqEPq5ZU ARi7w8Du…9RMf4yjv

752daf410ea81a…97311729592aba

1 in → 2 out1.0586 XPM227 B

AKmxBBzd…AHmu4uiE Aaxvxbud…mHnbUPuu

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.

2.099 × 10⁹⁴(95-digit number)

20991749435295031257…56161031267562742399

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

2.099 × 10⁹⁴(95-digit number)

20991749435295031257…56161031267562742399

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

2.099 × 10⁹⁴(95-digit number)

20991749435295031257…56161031267562742401

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

4.198 × 10⁹⁴(95-digit number)

41983498870590062514…12322062535125484799

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

4.198 × 10⁹⁴(95-digit number)

41983498870590062514…12322062535125484801

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

8.396 × 10⁹⁴(95-digit number)

83966997741180125029…24644125070250969599

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

8.396 × 10⁹⁴(95-digit number)

83966997741180125029…24644125070250969601

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

1.679 × 10⁹⁵(96-digit number)

16793399548236025005…49288250140501939199

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

1.679 × 10⁹⁵(96-digit number)

16793399548236025005…49288250140501939201

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

3.358 × 10⁹⁵(96-digit number)

33586799096472050011…98576500281003878399

Verify on FactorDB ↗Wolfram Alpha ↗

2^4 × origin + 1

3.358 × 10⁹⁵(96-digit number)

33586799096472050011…98576500281003878401

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