Block #213,161

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 5:41:21 PM · Difficulty 9.9204 · 6,589,391 confirmations

TWN

Bi-Twin Chain

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

Block Header

Block Hash

74746f09465c7da44bda0912b7b65c6855de168eeefa5d1f82a1631d05dbd878

View on Chainz ↗

Height

#213,161

Difficulty

9.920381

Transactions

Size

3.89 KB

Version

Bits

09eb9e16

Nonce

7,175

Timestamp

10/16/2013, 5:41:21 PM

Confirmations

6,589,391

Mined by

⛏️ P2SHAHeaxc341Ux7YKHViiDSbUuZHN7yt3yxEN

Merkle Root

31ffacd1256bc00155e79615c93864f406da250bbdaa24ac92e9cc4b717e4bb0

Transactions (4)

Coinbase3a63783b8fd792…897fb0c0e209aa

1 in → 1 out10.3200 XPM109 B

AHeaxc34…7yt3yxEN

6abfba466e3664…eb572a0ad0e205

9 in → 2 out692.5779 XPM1.38 KB

AMAYMEKe…9xu4iX9j AXYoqS2d…v1iM4SYB

912f03721acfbc…125027fa71c082

14 in → 2 out164.0100 XPM2.10 KB

AZyx7H4N…qnV3znHD AUw7jqNQ…njFKzkwd

001790d6ef5abc…60b2e67deb6d35

1 in → 2 out3.6874 XPM226 B

AZehgMKn…xPe3PbU3 AU8oQYoQ…CzMzYPjj

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.

8.147 × 10⁹⁴(95-digit number)

81478793947540339561…60016762011947140099

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

8.147 × 10⁹⁴(95-digit number)

81478793947540339561…60016762011947140099

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

8.147 × 10⁹⁴(95-digit number)

81478793947540339561…60016762011947140101

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

1.629 × 10⁹⁵(96-digit number)

16295758789508067912…20033524023894280199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

1.629 × 10⁹⁵(96-digit number)

16295758789508067912…20033524023894280201

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

3.259 × 10⁹⁵(96-digit number)

32591517579016135824…40067048047788560399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

3.259 × 10⁹⁵(96-digit number)

32591517579016135824…40067048047788560401

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

6.518 × 10⁹⁵(96-digit number)

65183035158032271649…80134096095577120799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

6.518 × 10⁹⁵(96-digit number)

65183035158032271649…80134096095577120801

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

1.303 × 10⁹⁶(97-digit number)

13036607031606454329…60268192191154241599

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 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

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

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